REPORT NO.

UCB/EERC-91/06

MAY 1991

PB93-114825

EARTHQUAKE ENGINEERING RESEARCH CENTER

COMPUTATION OFSPATIAllY VARYING GROUND MOTION ANDFOUNDATION-ROCK IMPEDANCE MATRICESFOR SEISMIC ANALYSIS OF ARCH DAMS

by

llPING ZHANG

ANIL K. CHOPRA

A Report on Research conducted underGrant No. CES-8719296 from the

, ...,.~NationalScience Foundation

COLLEGE OF ENGINEERING

UNIVERSITY OF CALIFORNIA AT BERKELEY

For sale by the National Technical Information

Service, U.S. Department of Commerce, Spring­

field, Virginia 22161

See back of report for up to date listing of EERC

reports.

DISCLAIMER

Any opinions, findings, and conclusions or

recommendations expressed in this publication

are those of the authors and do not necessarilyreflect the views of the National Science Foun­

dation or the Earthquake Engineering Research

Center, University of Califomia at Berkeley.

COMPUTATION OF SPATIALLY VARYING GROUND MOTION

AND FOUNDATION-ROCK IMPEDANCE MATRICES

FOR SEISMIC ANALYSIS OF ARCH DAMS

by

Liping Zhang

Anil K. Chopra

A Report on Research Conducted underGrant CES-8719296 from

the National Science Foundation

Report No. UCB/EERC 91/06Earthquake Engineering Research Center

University of CaliforniaBerkeley, California

May 1991

ABSTRACT'

This work on the development of analytical procedures to solve two problems

that are motivated by the earthquake analysis of arch dams is organized in two parts.

Presented in the first part is a direct boundary element method (BEM) to deter­

mine the three-dimensional seismic response of an infinitely-long canyon of arbitrary

but uniform cross-section cut in a homogeneous viscoelastic half-space. The seismic

excitation is represented by P, SV, SH or Rayleigh waves at arbitrary angles with

respect to the axis of the canyon. The accuracy of the procedure and implementing

computer program is demonstrated by comparison with previous solutions for the li­

miting case of two-dimensional response, recently obtained three-dimensional response

results for infinitely-long canyons, and three-dimensional boundary element method

solutions presented in this work for finite canyons. This procedure would enable ana­

lytical estimation of the spatial variation of ground motions around the canyon and

hence the possibility of exploring the effects of such variation on earthquake response

of arch dams.

In the second part, a direct boundary element procedure is presented to determine

the foundation impedance matrix defined at the nodal points on the dam-foundation

rock interface. The uniform cross-section of the infinitely-long canyon permits analy­

tical integration along the canyon axis leading to a series of two-dimensional boundary

problems involving Fourier transforms of the full-space Green's functions. Solution

of these two-dimensional boundary problems leads to a dynamic flexibility influence

matrix which is inverted to determine the impedance matrix. The accuracy of the pro­

cedure is demonstrated by comparison with previous solutions for a surface-supported,

square foundation and results obtained by a three-dimensional BEM for a foundation

of finite-width supported on an infinitely-long canyon. Compared with the three­

dimensional BEM, the present method requires less computer storage and is more

accurate and efficient. The foundation impedance matrix determined by this proce­

dure can be incorporated in a substructure method for earthquake analysis of arch

dams.

ACKNOWLEDGEMENTS

This research investigation was supported by the National Science Foundation

under Grant CES-8719296. The authors are grateful for this support.

Parts of this work were accomplished during the 1990 Fall semester while Anil

K. Chopra was on appointment as a Miller Research Professor in the Miller Institute

of Basic Research in Science, University of California at Berkeley.

ii

TABLE OF CONTENTS

ABSTRACT 1

ACKNOWLEDGEMENTS ii

TABLE OF CONTENTS iii

PART I: THREE-DIMENSIONAL ANALYSIS OF SPATIALLY VARY­

ING GROUND MOTIONS AROUND A UNIFORM CANYON IN A

HOMOGENEOUS HALF-SPACE , 1

INTRODUCTION . 3

SYSTEM CONSIDERED 6

FREE FIELD MOTION 8

BOUNDARY INTEGRAL FORMULATION 12

BOUNDARY ELEMENT FORMULATION 16

Discretization 16

Boundary Element Equations 18

Formation of System Matrices 19

Evaluation of Displacements 21

Summary '" , 21

DISCRETIZATION ERRORS 23

VERIFICATION 33

CONCLUSIONS . 41

REFERENCES 42

NOTATION 45

APPENDIX A: FREE-FIELD GROUND MOTIONS FOR SH, P, SV AND

RAYLEIGH WAVES 50

APPENDIX B: FULL-SPACE GREEN'S FUNCTIONS , 52

APPENDIX C: TRANSFORMED GREEN'S FUNCTIONS 53

Transformed Green's Displacement Functions 53

Transformed Green's Traction Functions 54

Expression for Matrix [C] 55

11l

PART II: IMPEDANCE FUNCTIONS FOR THREE-DIMENSIONAL

FOUNDATIONS SUPPORTED ON AN INFINITELY-LONG CANYON

OF UNIFORM CROSS-SECTION IN A HOMOGENEOUS HALF-SPACE

.............................. '" 57

INTRODUCTION 59

PROBLEM STATEMENT 60

FOURIER REPRESENTATION OF TRACTIONS AND DISPLACEMENTS 63

BOUNDARY INTEGRAL FORMULATION 65

BOUNDARY ELEMENT FORMULATION 68

Discretization 68

Boundary Element Equations 71

Formation of System Matrices 73

Evaluation of Displacements , 75

IMPEDANCE MATRIX 75

SUMMARY OF PROCEDURE 78

NUMERICAL ASPECTS 80

Selection of Sampling Points in Wavenumber Domain 80

Truncation and Discretization Errors 85

VERIFICATION AND COMPUTATIONAL COSTS 96

CONCLUSIONS 103

REFERENCES 105

NOTATION . 107

APPENDIX A: FOURIER TRANSFORM OF TWO-DIMENSIONAL SHAPE

FUNCTIONS 112

APPENDIX B: RELATION BETWEE~ {Uh AND {U}-k 116

APPENDIX C: RESTRICTIONS ON MESH LAYOUT ON f'c 119

IV

PART I

THREE-DIMENSIONAL ANALYSIS OF SPATIALLY VARY­

ING GROUND MOTIONS AROUND A UNIFORM CANYON

IN A HOMOGENEOUS HALF-SPACE

INTRODUCTION

Spatial variations in earthquake motions around the canyon have rarely been con­

sidered in practical earthquake analyses of arch dams, primarily for paucity of recorded

motions. Recent research studies have reported analyses of arch dam response to spa­

tially varying ground motion determined from two-dimensional wave scattering analysis

[1-3]. Three-dimensional analyses would obviously lead to more realistic prediction

of how earthquake motions vary over the canyon. This report is therefore concerned

with such analyses to determine the spatial variations in earthquake motions around

canyons supporting arch dams.

The spatial variation of the ground motion resulting from the scattering and

diffraction of waves by a canyon has been the subject of many studies. Most of these

studies are concerned with the two-dimensional response of infinitely-long canyons

of uniform cross-section. The anti-plane shear response of such geometrically two­

dimensional canyons to SH waves, parallel to the longitudinal axis of the canyon,

has been determined analytically for a semi-circular canyon [4], and a semi-elliptical

canyon [5]; and numerically for arbitrary cross-sections [6-8]. The numerical approaches

employed are based on the integral equation method [6], boundary method [7], and

finite element method [8].

The plane-strain response of geometrically two-dimensional canyons to P, SV and

Rayleigh waves normal to the canyon axis, which is much more complicated because of

mode conversions and coupled boundary conditions, does not seem to be amenable to

analytical solution. Consequently various numerical techniques have been developed.

Wong used a series of Lamb's solutions as trial function to represent the scattered field

and the complex amplitudes of the trial functions were determined through boundary

conditions in a least square sense by a generalized inverse method [9]. Sanchez-Sesma

et.al. applied Trefftz's wave function expansion method to construct the scattered

field [10]. The complex coefficients of the wave functions were then determined by

3

enforcing boundary conditions at both the canyon and the half-plane surface in a

least square sense. By expressing both the incident wave field and the scattered field

in terms of wave functions, Lee and Cao solved the diffraction of P and SV waves

at a two-dimensional circular canyon [11,12]. Their approach, however, results in a

set of infinite number of equations which can only be solved approximately. Various

boundary element methods (BEM), direct or indirect, have also been used to solve

the problem. Beskos et.al. analyzed the diffraction of waves by a trench using the

direct BEM with full-plane Green's functions [13]. An indirect BEM was applied

by Vogt et.al. to study wave scattering by arbitrary canyons in a layered half-plane

using half-space Green's function associated with distributed loads [14]. Nowak chose

the direct BEM with half-plane Green's functions associated with point loads [1]. By

combining the direct BEM with the discrete wavenumber Green's function, Kawase

investigated the time-domain response of a semi-circular canyon in a half-plane [15].

Hirose and Kitahara developed a special BEM to study the scattering of elastic

waves by inhomogeneous and anisotropic bodies in a half-space; the applicability of

their method, however, is limited because of the use of the fundamental solution of

elastostatics [16]. In addition to boundary methods mentioned above, various hybrid

methods (e.g. Mossessian and Dravinski [17]) have been used to study diffraction of

waves by surface irregularities.

Some of the numerical methods used for analysis of the plane-strain problem have

also been extended to study the diffraction of waves by finite-size, three-dimensional

surface irregularities including canyons and alluvial valleys. Wave function expansion

methods were applied by Sachez-Sesma et.al. [18,19], Eshraghi and Dravinski [20],

and Mossessian and Dravinski [21] to analyze the diffraction of waves by various finite

canyons and alluvial valleys. By expanding the incident and scattered fields in terms

of spherical wave functions, Lee analyzed the scattering of elastic plane waves by a

hemispherical canyon [22] and by a spherical cavity [23] in an elastic half-space. Using

the direct BEM, Banerjee et.al. studied wave barriers in multi-layered soil media [24]

4

and Rizzo et.al. investigated the scattering of waves by cavities in full-space [25,26].

The work on wave scattering and related problems has been reviewed by Wong [27],

Beskos [28] and Sanchez-Sesma [29].

Quasi two-dimensional formulations have been successful in analyzing three­

dimensional responses for systems with cylindrical geometry [30-34J. Wong et. al.

studied the three-dimensional dynamic response of infinitely-long pipelines by using

a cylindrical eigenfunction expansion technique [30]. The ground motion due to a

steady state dislocation propagating along a fault of finite-width and infinite-length

were solved by Luco and Anderson for a homogeneous half-space [31J and by Mendez

and Luco for a layered half-space [32] through integral representation theory. Liu

et.al. investigated the three-dimensional scattering of seismic waves by a cylindrical

alluvial valley embedded in a layered half-space by using a hybrid method where

the boundary integral representation and FEM are combined in such a way that the

problem associated with the singularity of Green's functions is eliminated [33].

The three-dimensional response of an infinitely-long, uniform canyon subjected to

waves at an arbitrary angle with respect to the axis of the canyon has been analyzed

recently by Luco, Wong and de Barros [34]. This problem was solved for a canyon

cut in a layered viscoelastic half-space by the indirect boundary method utilizing the

Green's functions associated with a concentrated load moving along the canyon axis.

The number and cross-sectional location of these moving sources are determined by

trial and error and their magnitudes are computed by imposing boundary conditions

along the canyon surface in a least square sense.

In this report, we solve the same wave diffraction problem by a canyon, except

that the half-space is homogeneous instead of layered and Rayleigh wave excitation

is also considered, by the direct boundary element method in conjunction with full­

space Green's function. The uniform cross-section of the infinitely-long canyon permits

closed-form integration along the canyon axis leading to a "two-dimensional" boundary

5

element problem involving Fourier transforms of the Green's functions. The assumption

of a homogeneous half-space seems to be appropriate for arch dam sites where similar

rock usually extends to considerable depth.

While we were working on this problem, an early, unpublished version of a paper

by Luco, Wong and de Barros [34J provided an independent set of results for evaluating

the accuracy of our results. In order to facilitate comparison, we follow their notation

and presentation in the next two sections and in the validation of numerical results.

SYSTEM CONSIDERED

The system considered consists of an infinitely long canyon of arbitrary but uniform

cross-section cut in a homogeneous viscoelastic half-space (Figure 1). The seismic

excitation is represented by P, SV, or SH waves incident on the canyon boundary

with arbitrary angles with respect to the vertical axis and the horizontal axes of the

canyon. Also considered are Rayleigh waves which approach the canyon horizontally

at an arbitrary angle with respect to the axis of the canyon. While appropriate

for exploratory purposes, free-field Rayleigh waves in a homogeneous half-space may

not be realistic since they imply larger amplitudes of the vertical component of the

wave than is supported by recorded data. Although the geometry of the canyon is

two-dimensional in that it is uniform along its length, the earthquake motion at the

canyon will vary along the axis of the canyon.

The viscoelastic half-space medium is characterized by the complex P and S wave

velocities Cp = cp(l + iTJp)I/2 and Cs = cs(1 + iTJs)I/2. The terms TJp and TJs represent the

constant hysteretic damping factors for P and S waves respectively; and cp and Cs

denote the P and S wave velocities if the medium were undamped. Similarly, the

Rayleigh wave velocity is given by Cr = cr(1 + iTJr )1/2 where Cr is the Rayleigh wave

velocity without damping and TJr is the corresponding hysteretic damping factor. Both

Cr and TJr are related to cp, cs, TJp and TJs.

6

Fig.! Infinitely-long canyon of arbitrary but uniform cross-section; rc is thecross-section of the canyon at x = 0; r i is the interface between thedam and foundation rock.

7

FREE FIELD MOTION

The first step is to determine the ground motion for free-field conditions, i.e. III

the half-space without the canyon. Considered first are the body waves. The seismic

excitation is represented by plane P, SV or SH waves in the underlying half-space

approaching from any direction. The normal to the wave front forms an angle 8v

with the vertical axis and an angle 8h with the axis of the canyon (Figure 2). The

incident motion in the underlying half-space, at any point xi = (x', y', z'), is described

by the three components of displacements given by

{U'(Xininc = A{u'}e-ik'x'+il/Z'+iwt (1)

where A is the amplitude of the incident displacement, w is the frequency of the

wave, k' = (wfcp)sin8v and v' = (wfcp)cos9v · for P-waves, and k' = (wfc s )sin8v and

v' = (wfcs)cos8v for S-waves. In equation (1), {u'} is the vector

for P-wavesfor SV-wavesfor SH-waves

(2)

The incident motion associated with Rayleigh surface waves, which propagate

hori- zontally, is given by

(3)

where A is a measure of the wave amplitude, k' = (wfc r ), and {u'(z')} is the vector

The free-field ground motion is the total motion in the half-space resulting from

the incident motion. The total motion due to surface waves is the same as the incident

motion (equation 3). In the case of body waves, the total motion is the incident

motion (equation 1) plus the reflected wave motion. The resulting total free-field

8

x

, x'~,, ,, ,, ,

, ', ', ", ,,~ffi, 7/oJ:

---F------:-,tr--~-_y, ,, ,, ,, ,,,,,,

z'__--~

r-Tir--'"""-----y------r-----~ X'

Rayleigh wave...

Fig.2 Canyon of uniform cross-section subjected to incident waves.

9

ground motion presented in Appendix A is well kn'own for the homogeneous half-space

[e.g. 35] but requires extensive calculation for a layered half-space [e.g. 36, 37]. In

either case, the free-field displacement field in the x'y'z'-coordinate system is denoted

by

(5)

where {u~fJ(z')} are the displacement components at the plane x' = O. In equation

(5), and in the sequel, the factor eiwt has been deleted for brevity. Similarly the

dependence of displacements and tractions on the frequency w is implied.

The free-field displacement is next transformed to the xyz-coordinate system in

which the boundary conditions at the surface of the canyon are imposed. For this

purpose, the rotation of coordinates is introduced:

(6)

The free-field displacements at any point x= (x, y, z) are

(7)

where {uJJ(io )} describes the displacements at i o = (0, y, z) on the y - z plane, k =(w/cp)sinOucos(h for P-waves, k = (w/cs)sinOucosOh for S-waves, and k = (w/cr)COSOh

for Rayleigh waves. Equation (7) suggests that waves travelling in the x'-direction

with speed c described by equation (5), may be interpreted as propagating in the

x-direction with an apparent speed of C/COSOh'

In the presence of the canyon, the total displacement field {u( i)} consists of the

free-field motion {uJJ(i)} and the scattered-field motion {us(i)}:

(8)

Similarly, the total traction vector {t (i)} on the canyon boundary and the half-space

surface is

(9)

10

where {tjj(xn and {ta(xn are the traction vectors associated with the free-field and

scattered field respectively, and n denotes the normal to the canyon and half-space

boundary at x.

The traction vector {tfJ(xn can be determined from the displacement field {ufJ(x)}

through the strain-displacement relations and material constitutive law. It can be

expressed as

(10)

The traction free condition on the boundary r c of the canyon requires that (using

equation 9)

(11)

On the boundary r h of the half-space, the same condition leads to

(12)

because the free-field solution satisfies the traction free condition.

To determine the total displacements, {u(x)}, around the boundary of the canyon,

only the scattered field in equation (8) needs to be evaluated since the free-field

displacements are known from equation (7). The scattered displacement field is of

the form

(13)

Thus, once the displacements along the boundary rein the plane x = 0 are determined,

their values at the dam-foundation rock interface are known from equation (13).

Thus, the problem reduces to evaluating the displacements along the canyon

boundary at the x = 0 plane associated with tractions at the canyon boundary and

half-space surface that are known from equations (11) and (12). This problem is solved

in the subsequent sections by the boundary element method, modified to recognize

the uniform cross-section and infinite length of the canyon.

11

BOUNDARY INTEGRAL FORMULATION

The boundary integral equation can be obtained from the reciprocal theorem

applied to a pair of elastodynamic equilibrium states; this theorem is a dynamic

extension of the classical Betti's law for elastostatic systems. Both the selected elas­

todynamic equilibrium states satisfy the wave equation over the half-space with a

cut canyon and radiation conditions at infinity, but not necessarily the boundary

conditions. The first state is the unknown scattered displacement field {us(x)} and

the associated tractions {t(x)} at the half-space surface and canyon boundary, which

can be determined from equations (11) and (12). The second state is defined by the

displacements {u*(x,xos)} and tractions (t*(x,xos)} in a full-space due to concentrated

forces {P} = {Px , Py, Pz}T applied at a point xos = (0, Ys, zs) on f' c or f'h, the intersection

of the system boundary and the x = 0 plane. According to the reciprocal theorem,

the work done by the tractions of the second state in undergoing the displacements

of the first state is equal to the work done by the tractions of the first state through

the displacements of the second state:

The integration is over the entire boundary S consisting of the surface of the half­

space and the boundary of the canyon, modified slightly to avoid the singularity at

the point of load application (Figure 3).

nThe displacements {u*(x, xos)} and tractions {t*(x, xos)} at a "receiver" point x =

(x, y, z) on the boundary S are related to the concentrated forces applied at the

"source" point xos = (0, Ys, zs) on the boundary f' c and f\ :

(lSa)

(lSb)

The traction vector is associated with the normal n to the boundary, and [G*(x, xos)]n

and [P'(X, xos)] are 3 x 3 matrices of Green's functions. The first, second and third

12

Fig.3 Definition of boundaries and source point; r c is the cross-section ofthe canyon at x = 0, rh is the x = 0 line on the surface of the half­space, and r. is the small contour of radius r. around the sourcepoint.

13

ncolumns of the matrices [G*(x, xos)] and [F*(i, xos)] 'correspond to the displacement and

traction vectors at x for a unit point load acting at xos in the x, y and z directions,

respectively. The 3 x 1 vector {P} represents the unknown amplitudes of the forces.n

Expressions for matrices [G*(x, xos)] and [F*(x, xos)] are presented in Appendix B [38].

The boundary integral equation is obtained by substituting equation (15) into

equation (14) and recognizing that the latter should be satisfied for all force vectors

{P}=

[ /00 [F*(x,xos)f{us(i)}dxdf =Jrcur"ur. -00~ _/00 [G*(i, ios)f{t(i)}dxdf

Jrcur"ur. -00 (16)

For each source point xos, this equation represents three scalar equations in the

unknown displacements {us(i)}.

Equation (16) can be simplified by substituting equations (10), (11) and (13) and

analytically evaluating the integral along the x-axis, leading to

(17)

(18b)

(18a)

where

[G(io, xos)] = 1:[G*(x, xos)]e-ikXdx

[F(xo, Xos)] = 1:[F*(x, xos)]e-ikxdx

Expressions for these Fourier transforms are given in Appendix C. Thus the original

boundary integral equation (16) has been transformed to a simpler form (equation

17) in which the remaining integration is only along the curve l'c u l'h U 1's at the

x =0 cross-section of the system.

The final step is to make the radius rs of the small arc 1's around the source

point, ios, approach zero. Following [39], it can be shown that

(19)

14

(21)

where

[C( ios)] = lim [[F( i o, ios)fdf (20)r.-olr.

A general expreSSIOn for the 3 x 3 matrix [C(ios)), which depends on the geometry

of the boundary at the source point, is presented in Appendix C. If the boundary is

smooth, i.e., it has a unique tangent, at the source point, Cij(ios) =Cij/2.

The final form of the boundary integral equation is obtained from equation (17)

after utilizing equation (19):

[C(ios)]{us(ios)} + [ [F(io, ios)]T{us(io)}dflreur"

= - ~ _ [G(io, xos)]T{fff(io)}dflrc ur"

nIn this equation, the traction vector {t ff (i o)} is known from equation (10) and the

ntransformed Green's function matrices [G(io,ios)] and [F(io,xos)] are known from

equation (18) and Appendix C, whereas the scattered-field displacements at the plane

x = 0 are the unknowns. These include {us(ios)} at the source point and {us(xo )} at

the rest of the boundary feu l'h. Once the displacements along the boundary feU f h

are determined, the displacements at the rest of the boundary f, and in particular

at the dam-foundation rock interface can be determined from equation (13).

By taking advantage of the uniform geometry of the system along the x-direction,

the three-dimensional boundary integral equation (16) has been reduced to a "two­

dimensional" problem (equation 21) involving Fourier transforms of Green's functions.

This was possible because the x-variation of tractions and displacements is known

analytically (equations 11 and 13) which permitted closed-form integration along the

x-axis. The reduced problem is really not two-dimensional because the unknown

scattered displacements along the boundary feU f h are three-dimensional vectors.

If the incident waves are perpendicular to the x-axis, Oh = 900 and therefore k =0,

equation (18) becomes

(22a)

15

(22b)

nIn the 3x3 matrices [Go(xo,xos)] and [Fo(xo,xos)], the (1,1) entry represents the Green's

functions for the 2-D anti-plane shear motion; the (2,2), (2,3), (3,2) and (3,3) entries are

the Green's functions associated with the 2-D plane-strain motion, and the remaining

entries are zero because the in-plane and anti-plane motions are uncoupled. In this

case, the boundary integral equation governs two uncoupled problems: the anti-plane

shear problem associated with an incident SH wave and the plane-strain problem

associated with incident P, SV and Rayleigh waves.

BOUNDARY ELEMENT FORMULATION

Discretization

In its present form, equation (21), which represents the exact formulation of

the problem, cannot be solved analytically to determine the scattered displacements,

even for the simplest canyon geometry. Therefore, the integration domain f'c u f' h is

discretized into M one-dimensional elements (Figure 4). Obviously, the discretization

extends to only a finite distance along f' h on the half-space surface, which should be

large enough to provide accurate solutions. In the simplest form, each element has

two nodes, and the interpolation functions are linear. Following isoparametric element

concepts, these interpolation functions are Nt (() = 0.5(1 - () and N 2(() = 0.5(1 +()where ( varies from -1 to 1 (Figure 4). Thus the discretized system with M elements

has M n = M +1 nodes.

The variation of the scattered-field displacements over the jth element can be

expressed in terms of nodal displacements

where [N(()] is a 3 x 6 matrix consisting of interpolation functions

(23)

(

Nt[N(()] = ~ (24)

16

FhelementJ

o

z

y

Fig.4 Boundary element discretization.

17

and {iT}; is the vector of displacements at the two nodes - j and j + 1 - of the

element

Boundary Element Equations

The boundary integrations in equation (21) can now be expressed as summation

of integrals over all the M elements

M

[C(xos)]{us(xos)} +2:=i [F(Xo,Xos)]T{us(xo)};df;=1 t j

(25)

where 1"; represents the extent of the jth element. The summation on the right side

extends over only the elements of fe, the canyon boundary (Figure 4), because the free­

field tractions {tff(xo )} vanish on f h , the half-space surface, and hence the contribution

of the corresponding elements is zero. By expressing the displacement field in terms

of nodal displacements through equation (23), and changing the integration variable

to (, the contributions of the jth element in equation (25) become

[ [F(xo,xos)];{us(xo)};df = ~Ijjl [F«(,xos)]J[N«()]d({iT}j (26a)itj -1

[ [G(xo, xos)];{tfJ(xo)}jdf = -211j /1 [G«(, xos)];{tfJ«()}jd( (26b)~j -1

where I; is the length of the jth element. Substituting these equations, equation (25)

can be expressed in matrix form:

Equation (27) represents three equations associated with the source point xos

(0, Ys, zs), where the 3 X 3 matrix [C(xos)] is given by equation (20), the 3 x 1 column

vector {us(xos)} represents the unknown displacements at the source point, {iT} is a

18

3Mn x 1 column vector of unknown nodal displacements (3 displacement components

at each of the M n nodes), [A(xos)] is a 3 x 3Mn matrix assembled from the integrals of

the products of the transformed traction Green's functions and element interpolation

functions:- 1

[A(xos)] = 2M 1

~ I j 11 (F«(,xos)]J[N«()]d(a.s.semble

(28)

and {Q(ios)} is a 3 x 1 vector defined by

(29)

The matrix [A(xos)] is obtained by appropriately assembling the contribution of the

various elements, a process that is reminiscent of the stiffness matrix assembly process

in the direct stiffness method. The vector {Q(xos)} is obtained by direct addition of

the contributions of the various elements. The construction of both of these matrices

for a small system is illustrated in Figure 5.

Equation (27) is a discrete form of equation (21) obtained by introducing two

approximations: (1) interpolation of displacements over each element in terms of nodal

displacements (in this particular formulation the interpolation functions were chosen

to be linear); and (2) truncation of the integration over f h to a finite distance.

Formation of System Matrices

Associated with the source point xos = (0, Ys, zs), equation (27) represents 3 linear

algebraic equations in 3Mn unknown displacements. Obviously a total of 3Mn equations

are needed to determine the unknown displacements. These equations can be obtained

by successively choosing the source point at each of the nodes of the discretized

boundary. When the source point coincides with the ith node, Xi, equation (27)

becomes

\.

(30)

Such equations associated with all nodes can be obtained from equation (30) by

19

•5

++++ooofoofoofoo~ ••~ ••

ooofoofoofoo~··r··000000000000 ••••••

•

}{U}

•~+~+t:+~+~

contribution ofelement 3 = {Q(xos)h

...~..~..~..~..~..···~··~··I···~··~··••••••••••••••••••

Fig.5 Assembly of Matrices.

20

varying i = 1,2, ... , M n and assembling the resulting equations in the following form:

[AHU} = {Q}

where [A] is a square matrix of order 3Mn given by

(31)

(

[C(id]

[A] = (32)

(33)

and {Q} is a 3Mn x 1 column vector

{Q} = ( {Q(Xl)) 1{Q(iMn )}

The coefficient matrix [A] is unsymmetric and fully populated, which is typical of the

boundary element method equations.

Evaluation of Displacements

Solution of the system of linear algebraic equations (31) leads to the nodal dis­

placements {U} and thereafter the displacement variation over each finite element from

equation (23). The resulting displacements represent an approximation to {u,,(io )},

the scattered displacement field along the boundary f'c u f' h at the plane x = 0. The

scattered displacements {u,,(in on the boundary surface at any other plane are then

given by equation (13), which when combined with the known free-field displacements

(equation 7) lead to the total displacements, {u(x)} (equation 8).

Summary

The boundary element procedure developed m the preceding sections may be

summarized as follows:

1. Discretize f'c, the canyon boundary at x = 0, and f'h, the half-space surface at

x = 0, into M line elements. If linear interpolation functions are selected for each

element, the discretized system contains Mn = M +1 nodes.

21

2. Establish equation (30) which includes three linear algebraic equations associated

with the ith node, X;, in 3Mn unknowns (3 displacement components at each of

the M n nodes) by the following steps:

(a) Compute the elements of the 3x3 matrix [C(x;)] which depend on the geometry

of the boundary at the ith node, see Appendix C. These elements are given

by Cjk(xd =Ojk/2 if the boundary is smooth, i.e., it has a unique tangent at

X;.

(b) Compute the 3x3Mn matrix [..4(i;)] from equation (28) by assembling thecontri­

butions of all the M elements, as described in Figure 5. Element contributions

are determined from the integrals of the transformed traction Green's functions

(equation I8b) and element interpolation functions. Analytical expressions for

the transformed traction Green's functions are available in Appendix C.

(c) Compute the 3 x 1 vector {Q(id} from equation (29) by adding the contri­

butions of the elements on the canyon boundary, as described in Figure 5.

Element contributions are determined from the integrals of the products of

transformed displacement Green's functions (equation 18a) and free-field trac­

tions. Analytical expressions for the transformed Green's functions are avail­

able in Appendix C and the traction vector is determined from the free-field

displacements {uff (i)} given by equation (7) through the strain-displacement

relations and material constitutive law.

3. Repeat the computations summarized in step 2 for each of the nodes, i = 1,2, ... , M n

to determine [C(i;)], [..4(i;)] and {Q(i;)} for all the nodes.

4. Evaluate the square matrix [A] of order 3Mn and vector {Q} of size 3Mn for the

boundary element system from the corresponding individual nodal matrices (step

3) using equations (32) and (33).

5. Solve the system of 3Mn linear algebraic equations to determine the nodal dis­

placements {U} and thereafter the displacement variations over each finite element

22

from equation (23).

6. Determine the scattered displacements {u s ( x)} at the dam-foundation rock inter­

face, or any plane other than x = 0, from the displacements at the x = 0 plane

(step 5) using equation (13).

7. Add the scattered displacements {us(x)} from step 6 and the free-field displace­

ments {uff(x)} , determined from equation (7), to obtain the total displacements

{u(x)}.

A computer program has been developed to implement the procedures summarized

above for an incident wave of frequency w with its propagation direction defined by

angles (Jv to the vertical, z-axis and (Jh to the horizontal, x-axis of the canyon.

The boundary element formulation summarized above is for a canyon cut m a

homogeneous viscoelastic half-space. In principle, this method can be extended to a

layered half-space with non-horizontal layers. A boundary integral equation, similar

to equation (21), can be obtained for each layer and the underlying half-space. The

interfaces between layers would also require discretization because full-space Green's

functions are used. Thus this approach may become computationally impractical for

multi-layered systems.

DISCRETIZATION ERRORS

As mentioned earlier, discretization of the boundary obviously extends only a finite

distance, Lh' along the half-space surface at the plane x = 0 (Figure 4). The minimum

Lh necessary for accurate solution of the problem is examined next. Computed by

the present procedure, with different L h values, the displacement amplitudes around

a semi-circular canyon of radius L are presented in Figures 6 to 9 for P, SH, SV

and Rayleigh waves for incidence angles fh = 30° and (Jv = 45° and three values of

the normalized frequency, n =wL/'lrcs • For L =800 ft, and Cs =6000 ft/sec, n =0.5,1

and 2 represent frequencies f = 1.9,3.8 and 7.5 Hz. The results are presented for four

selections of Lh : L, 2L, 4L and 6L; in each case the element size is taken as L/8. The

23

4. = 1L4. = 2L

- - - 4. = 4L4. = 6L

3 --.--------..,o = 0.5..

.2 2Q)"0~

~Q.E 1<.Q.m

C 0 -+---r---r-"""T""""""T"""....-....--.---i

3 .........---------,o = 0.5

o = 1.0 o = 2.0

o = 2.0

o = 0.5 o = 2.0o = 1.0

.Q.m

C 0 -+---'---'-"""T"""--r---r---;---;--;

3--.---------.,

.""

.2 2Q)"0~

.~Q.E 1<

..

..:! 2Q)"0~

~Q.E 1<

-1 0 1Distance y/L

2 -2 -1 0 1Distance y/L

2 -2 -1 a 1Distance y/L

2

Fig.6 Influence of Lh on computed response of a semi-circular canyon toP-wave excitation (fh = 30°, ()v = 45°).

24

4t = 1LL,. = 2L

- - - L,. = 4LL,. = 6L

2

{] = 2.0

{] = 2.0

-1 0 1Distance y/L

2 -2

{] = 1.0

{] = 1.0

{] = 1.0

-1 0 1Distance y/L

2 -2

0.5

-1 0 1Distance y/L

(] = 0.5

{] = 0.5

3 ........----------,

Q.encO-+---.---.---.---,.--,.~___.--I

3 ........----------,

-;..::J- 2enQ)1J::J

:e=.

E"1<

.CiencO-+--.---r--r----,----r--...,.---r~

-2

CiencO-+---.---.---,.~___.___,,...-r_l

3 -,--------...,

..::J- 2enQ)1J::J~

E"1<

..::J- 2enQ)

1J::J~

E"1<

Fig.7 Influence of Lh on computed response of a semi-circular canyon toSH-wave excitation (fh = 30°, Bv = 45°).

25

Fig.8 Influence of Lh on computed response of a semi-circular canyon toSV-wave excitation (lh = 30°, Bv = 45°).

26

------------ Lh = 1L - - - Lh = 4L- - -- ~ = 2L ~ = 6L

30 = 0.5 0 = 1.0 0 = 2.0..

:l2

enII)"0:l~

E1<.0-enC 0

30 = 0.5

-...:l

2enQ)

"0:l~

Q.E<Q.enC 0

40 = 0.5 0 = 1.0 0 = 2.0

..;3 /h\en / ", ,

II)I' ,

"0 2 II ,,~,

:l I

-.J

'~" AQ.E ~., , ,< .......

Q.en

CS 0-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2

Distance y/L Distance y/L Distance y/L

Fig.9 Influence of L h on computed response of a semi-circular canyon toRayleigh wave excitation (fh = 30°).

27

semi-circular curve fe, which is the intersection' of the canyon and plane x = 0, IS

discretized into 24 elements.

It is apparent from the results of Figures 6 to 9 that, in most cases, the displace­

ments on the canyon surface are not very sensitive to the Lh value. The dependence

on Lh decreases for higher frequency. It is more meaningful to define the required Lh

as a multiple of the shear wavelength. Such results demonstrate that if Lh exceeds

twice the shear wavelength the results are accurate.

The dependence of the boundary element solution on the fineness of the dis­

cretization is examined next. Computed by the present procedure, the displacement

amplitudes around a semi-circular canyon are presented in Figures 10 to 13 for P,

SH, SV and Rayleigh waves for normalized frequency n = 0.5,1,2 and (Jh = (Jv = 45°.

The corresponding shear wave lengths are equal to 4L, 2L and L where L is the

radius of the canyon. The canyon boundary is divided into N e elements and the

half-space surface into N h elements with the latter covering an extent of L h = 4L on

both sides of the canyon. The element size on f h is about the same as that on f e •

The results obtained from three different discretizations are shown in Figures 10 to

13 together with what might be considered as the "exact" result. The latter was

obtained with very fine discretization, N e = 32 and N h = 40. It is apparent that, as

the discretization becomes finer, the results converge to the "exact" values. A coarse

mesh gives reasonably accurate results only at low frequency, and a finer mesh must

be used at high frequency to describe the rapid variations of the displacements. It

is the ratio of the shear wavelength to the element size that determines if a chosen

mesh is fine enough. Reliable results are obtained when this ratio is greater than 5.

In summary, the present method is reliable and can be used to analyze the spatial

variations of motion around an infinitely-long canyon of arbitrary shape. In order to

control discretization errors, the element size on the canyon surface should be kept

smaller than one-fifth of the shear wavelength, and the half-space surface should be

28

o Nc=4, Nh=5Nc=8, Nh =10

Nc= 16. Nh=20-- Nc=32. Nh=40

3--.------------,o = 0.5..

..2 2CD'0:J

;'=Q.E 1<Q.en

° O-+----r---r---r---r---r---.-.---l

3---r----------,o = 0.5

o = 1.0

o

o = 1.0

o = 2.0o

o

o = 2.0

o = 1.0o = 0.5

-,..:J 2CD'0:J

;'=Q.E 1<Q.en

°O-+---.-r---r--..,...--r---r----T_

3--.------------,

..:J 2CD'0:J

.":a.E 1<

2-1 0 1Distance y/L

2 -2-1 0 1Distance y/L

2 -2-1 0 1Distance y/L

Q.en

°O-+---r----r---r-....---,r--r--,.........-l

-2

Fig.l0 Influence of mesh refinement on computed response of a semi-circularcanyon to P-wave excitation ((h = (Jv = 45°, Lh = 4L).

29

o Nc=4, N..=5Nc=8, N..=10

Nc= 16, N..=20-- Nc=32, N,,=40

4--.---------,o = 0.5 o = 1.0 o = 2.0

·3:Jo

o

CD"0:J 2

;0=Q.E< 1.Q.(I)

is o-+-,--,--,-...,.--r-...-"""'"4--.----------.,

o = 0.5 o = 1.0 o = 2.0

o = 0.5

2

o = 2.0

o

-1 0 1Distance y/L

2 -2

o

o = 1.0

-1 0 1Distance y/L

2 -2-1 0 1Distance y/L

- .. 3..2CD"0:J 2

;0=Q.E<1.Q.(I)

is 0-+-,--,--,-...,..........,.....................-13--r---------,

Q.(I)

iSO-+-.----,.l~__,.___r--,___,,..._'

-2

..:J 2CD"0:J

;0=Q.E 1<

Fig.ll Influence of mesh refinement on computed response of a semi-circularcanyon to SH-wave excitation (Oh = Ov = 45°, Lh = 4L).

30

o Nc=4, Nh=5Nc=8, Nh=10

- - Nc=16. Nh=20-- Nc=32. Nh=40

3--r----------,a = 0.5 a = 1.0 a = 2.0

2

o

o

o

a = 2.0

a = 2.0

o

o

o

2 -2 -1 0 1Distance y/L

a = 1.0

2 -2 -1 a 1Distance y/L

..:)

2GJ

'U 0:);0=Q.E 1<a.(I)

i5 a3

a = 0.5

-,..,:)

2GJ'U:)~.-Q.E 1<a.(I)

i5 a4

a = 0.5

.. 3:)

GJ'U:) 2~

Q.E<a.(I)

c a-2 -1 a 1

Distance y/L

Fig.12 Influence of mesh refinement on computed response of a semi-circularcanyon to SV-wave excitation (fh = f)v = 45°, Lh = 4L).

31

Fig.13 Influence of mesh refinement on computed response of a semi-circularcanyon to Rayleigh wave excitation (fh = 45°, .Lh = 4L).

32

discretized on both sides of the canyon over a distance of at least two times the shear

wavelength.

VERIFICATION

The analytical procedure and the implementing computer program were tested by

solving problems for which previous results are available. In all cases, an infinitely­

long canyon with semi-circular cross-section of radius L cut in a homogeneous half­

space is considered. The earlier results [4,9] for the two-dimensional response of the

canyon are for a elastic half-space with cp = 2cs , i.e. Poisson's ratio = 1/3. The

recent results [34] for the three-dimensional response of the canyon are for a slightly

dissipative viscoelastic half-space characterized by cp = 2cs and TIp = TIs = 0.02. The

present approach is implemented for the latter set of material properties, with the

canyon boundary f'c discretized into 32 equal elements and each side of the half-space

boundary f'p is discretized into 30 equal elements covering Lh =4L (Figure 4).

The results obtained by the present method for the two-dimensional anti-plane

shear motion arising from an SH wave of unit amplitude impinging normal to the

canyon axis are compared in Figure 14a with the exact solution of Trifunac [4]. The

results presented describe the displacement amplitudes around the canyon for Ov =

1°,45° and 85° and a dimensionless frequency n = wL/1rcs = 1. A similar comparison

is presented in Figure 14b for Ov = 0° and Oh = 0° which corresponds to a vertically

incident SH wave with particle motion perpendicular to the axis of the canyon. For

this plane-strain case, the numerical results are available (e.g. Wong [9]). Figure 15

shows the results for the two-dimensional response of the canyon to P and SV waves

impinging normal to the axis of the canyon (Oh = 90°) at various vertical angles Ov

and for a dimensionless frequency n = wL/1rcs = 1. The comparisons in Figures 14

and 15 indicate that the present method provides reasonably accurate results for the

two-dimensional limiting cases considered.

We now turn to the three-dimensional response of a canyon to incident waves

33

4--,-----------------------~

3

-.::l

V"U::l 2~

0.E«

0.1enC5

?9

r:I/

o b jJ'0

17., = 1°- - - - 17., = 45°--------- 17., = 85°

1.00.50.0Distance y/L

-0.5O-+----r----r---,-,...--.--....,---,-~____r__,_-r__;____r__,__.___r_,__,-r_i

-1.0

4 """""T"""--------------------------,----Iurl- - - - IUrl

3

2oDistance y/L

-1

II

90'D-tJ g,.O.9

O-+---..-.....--..,..--...,.....--r---,.-.,..---.--4It--.-...---.--..---,.-,.......-.....--r----r---1

-2

0.en

C5

Fig.14 Comparison of results of present method (lines) with previousresults (Ref. [4] shown by 0, and Ref. [9] shown by c ) for SH­wave excitation; n = 1.

34

P-waves SV-WQves

3-,---------------. 4---r------------~

,'O°'Qoa> ,I \I \

3.....,....---------------.~ =300

I .,....I

I

i 2JJ1Ci

E 0« 1\'Ci1~

i5 J

4-,-------------~

3

2

3-,-------------, 10 """T""-------------..

2-1 0Distance y/l

2

6

4

8

O+-,.,........_r...,....,~T"'""T""...._r...,....,~l,__r_"'T""'T_,_\

2 -2I

-1 0Distance y/l

o I

-2

Fig.IS Comparison of results or present method (lines) with previousresults (Ref.[9J shown by 0) for P- and SV- wave:! excitations;n = 1, ()h = 900

•

35

impinging from an arbitrary direction. The calculated amplitudes of the three dis­

placement components (lu,,;I,luyl, and luzl) for incident P, SH and SV waves of unit

amplitude, dimensionless frequency n =wL/1rcs =1, vertical angle of incidence Ov =45°,

and horizontal angle of incidence Oh = 45° are presented as solid lines in Figure 16

for an infinitely-long canyon of semi-circular cross-section, dashed lines represent the

results obtained by the indirect boundary method [34]. This comparison indicates that

the present method provides reasonably accurate results also for the three-dimensional

problem.

Also included In Figure 16 are the displacement amplitudes around the canyon

computed by analyzing a finite-length canyon by a fully three-dimensional verSIOn

of the present boundary element method. In this case, discretization is necessary

even along the canyon axis (Figure 17) in contrast to the method presented in the

preceding sections where integrations in the x-direction could be evaluated analytically

(equation 14). The results at x = 0 obtained for the finite-length canyon of Figure 17

with d =4L and Lh = 1.5L are shown in Figure 16 by discrete symbols. The finiteness

of the canyon has only a small effect on the displacements far from the ends of the

canyon. The results for the finite-length canyon can be improved in accuracy at the

expense of increased computational effort - by increasing Lh and by using a finer

discretization on the canyon boundary.

The results presented in Figure 18 are for an infinitely-long canyon (Figure 1) and

finite canyons (Figure 19) of four different lengths subjected to Rayleigh waves with

dimensionless frequency n = wL/1rcs = 1, and horizontal angles of incidence 0h = 30°

and 60°. This comparison indicates that the results obtained for infinite and finite

canyons are generally similar and become closer with increasing length of the finite

canyon. The accuracy of the results obtained for the finite-length canyon is limited

because of the coarseness of the discretization on the canyon boundary and the short

distance, Lh = 1.5L, over which the discretization extends on the half-space surface

(Figure 19). Compared with the fully three-dimensional version of the boundary

36

2

SV-woves

-1 0 1Distance y/L

2 -2

o

SH-waves

o

2 -2 -1 0 1Distance y/L

-2

P-waves

2

2

o -+-----,I,---,-~_,__,_-.-.....,

-1 0 1Distance y/L

O--+--r---r---r---r---r---r---r--l

3--r-----------,

2

3-,------------,

O-+-~_r__,____r_,___r__,--1

3-.------------.,

Fig.16 Comparison of results of present method (continuous lines) with previ­ous results (Ref.[34] shown by dashed lines) for P-, SH- and SV-waveexcitations (n = 1.0, (h = ()v = 45°, "7P = "7s = 0.02). Results forfinite-length canyon (d = 4L) are shown by discrete symbols.

37

z

x

x

l

d

l

Fig.17 Canyon of finite-length.

38

y

2

3~-------------.,

•;lF.....I

o !

0 t 0

3 J 317h=30o "h=60;

IJ

i2-1 2

IUyl I1 •

c• 0c0

2-1 aDistance y/L

O-+...---.~...___."""'T_"...___.__r_T""""T_,_T""""T_,_,._,.....-rl

-2

2

4 -T'"----------------,

3

2

•8

8

-1 0Distance y/L

o i

-2

4.....,....--------------,J 19h=30°

I

3..JIi

....i

Fig.IS Comparison of results for infinite canyon by the present method (lines)and for finite canyons of different lengths (d = L shown by 0, d = 2Lby c , d = 3L by [> and d = 4L by *) by the 3-D BEM for Rayleighwave excitation (0 = 1, 77p = 77. = 0.02).

39

x

..J

..Jas@)

ECDECD

Q)

EJcII

"'C

L 1.5L

n m a1 4 0.2502 6 0.3333 8 0.3754 8 0.500

y

Fig.19 Top view of a quarter of the discretized system for finite canyons offour different lengths.

40

element method, the present approach is much more efficient as it permits analytical

integration along the canyon axis.

CONCLUSIONS

A direct boundary element procedure to calculate the three-dimensional response of

an infinitely-long canyon of arbitrary but uniform cross-section cut in a homogeneous

viscoelastic half-space to incident seismic waves has been presented. The seismic

excitation is represented by P, SH, SV or Rayleigh waves at arbitrary angles with

respect to the axis of the canyon. The uniform cross-section of the infinitely-long

canyon permits analytical integration along the canyon axis of the three-dimensional

boundary integral equation. Thus the original three-dimensional problem is reduced

to a computationally efficient "two-dimensional" boundary element problem involving

Fourier transforms of full-space Green's functions.

The accuracy of the procedure and implementing computer program has been ver­

ified by comparison with previous solutions for the limiting case of two-dimensional re­

sponse, recently obtained three-dimensional response results for infinitely-long canyons

by the indirect boundary method, and three-dimensional boundary element method

solutions presented here for finite canyons. The finiteness of the canyon has only a

small effect on the displacements far from the ends of the canyon. Compared with the

fully three-dimensional version of the boundary element method, the present method

is much more efficient as it permits analytical integration along the canyon axis. This

method is suitable for research studies concerning the influence of spatial variations

in input motions on the earthquake response of arch dams.

41

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42

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28. D.E. Beskos, "Boundary element methods in dynamic analysis," Applied MechanicsReviews, 40, No.1, pp.1-23, 1987.

29. F.J. Sanchez-Sesma, "Site effects on strong ground motion," Soil Dynamics andEarthquake Engineering, 6, No.2, pp.124-132, 1987.

30. K.C. Won~, S.K. Datta and A.H. Shah, "Three dimensional motion of a buriedpipeline. I. analysis," J. Engineering Mechanics, ASCE, 112, No. 12, pp.1319-1337,1986.

31. J.E. Luco and J.G. Anderson, "Steady-state response of an elastic half-space toa moving dislocation of finite width," Bull. Seism. Soc. Am., 73, pp.1-22, 1983.

32. A. Mendez and J.E. Luco, "Near-source ground motion from a steady-state dislo­cation model in a layered half-space," J. Geophysical Research, 93, pp.12041-12054,1988.

33. S.W. Liu, S.K. Datta, M. Bouden and A.H. Shah, "Scattering of obliquely incidentseismic waves by a cylindrical valley in a layered half-space," Earthquake Engineeringand Structural Dynamics, to be published.

34. J.E. Luco, H.L. Wong and F.C.P. de Barros, "Three-dimensional response of acylindrical canyon in a layered half-space," Earthquake Engineering and StructuralDynamics, 19, pp.799-818, 1990.

35. K.F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, 1975.

43

36. J.E. Luco and H.L. Wong, "Seismic Response of Foundations Embedded in aLayered Half-Space," Earthquake Engineering and Structural Dynamics, 15, pp.233­247, 1987.

37. J.P. Wolf and P. Obernhuber, "Free-Field Response from Inclined SV- And P­Waves And Rayleigh-Waves," Earthquake Engineering and Structural Dynamics, 10,pp.847-869, 1982.

38. A.C. Eringen and E.S. Suhubi, Elastodynamics, Vol. II, Linear Theory, AcademicPress, N.Y., 1975.

39. F. Hartmann, "Computing the C-matrix in non-smooth boundary points," New De­velopments in Boundary Element Methods, (Ed. by C.A.Brebbia), CML Publications,pp.367-379, 1980.

44

a

A

[Aj

[AJ

b

cp

[C(x)J

d

f

[FJ

NOTATION

numerical coefficient

= ..jk2 - k;= Jk2 - k;measure of incident wave amplitude

square coefficient matrix for the boundary element system (equation32)

3 x 3Mn matrix assembled from the integrals of the products of[FjT and [NJ associated with a single source point (equation 28)numerical coefficients

amplitude of reflected P wave

amplitude of reflected SV wave

numerical constant

= k r (1 - c; / c;)1/2

= kr (1- c;/C~Y/2

P wave velocity including damping

P wave velocity without damping

Rayleigh wave velocity including damping

Rayleigh wave velocity without damping

shear wave velocity including damping

shear wave velocity without damping

3 x 3 matrix related to the geometric shape of the boundary at x(equation 20)

= vi(x - x.)2 + (y - y.)2 + (z - Z.)2, distance between source and re­ceiver points

= J (Y - y.)2 + (z - z.)2, distance between source and receiver pointsprojected on plane x = 0

frequency in Hz

transform€ j full-space Green's traction functions (equation 18b)

45

[F*]

[GJ

k

k'

Ii

L

M

[NJjNi

traction functions associated with full-space Green's displacementfunctions [G*]traction functions associated with the transformed full-spaceGreen's displacement functions [F] for k = a (equation 22b)transformed full-space Green's displacement functions (equation18a)full-space Green's displacement functions

transformed full-space Green's displacement functions for k = a(equation 22a)

=yCI

wavenumber in x-direction

wavenumber in x'-direction

P wavenumber

Rayleigh wavenumber

shear wavenumber

modified Bessel functions of second kind of order zero and orderone

length of the jth element on I'c U I'h

half width of the uniform canyon

discretization range on the half-space surface

number of one-dimensional elements on the boundaries I'c U I'h

number of nodes on the boundaries I'c U I'h

element number for the first and the last one-dimensional elementson boundary l'coutgoing normal to the boundary and its components

shape function matrix for the one-dimensional element on f'c U f'h;

jth shape function

number of elements on canyon boundary l'c

number of elements on one side of the half-space boundary f'h

vector and components of concentrated forces associated with thesecond elastodynamic state

46

{Q}

r.

s

t

{t}

{l}

{t*}

{tf! }

{If f}

{to }

[T]

{u}

{u}

{U/}

{U/}

{u*}

{Uff}

{uff }

{u~ff }

{u. }

{u.}

vector assembled from {Q} associated with all nodes on i'c u f\(equation 33)

3 x 1 vector from integrals of the products of transformed Green'sdisplacement functions and the free-field tractions (equation 29)radius of a circular curve around a source point

the entire boundary of the system which consists of the surface ofthe half-space and the boundary of the canyontime

traction vector

traction vector at plane x = 0

traction vector associated with the second elastodynamic state inreciprocal theoremtraction vector associated with the free-field motion

traction vector associated with the free-field motion at plane x =0

traction vector associated with the scattered field motion

3 x 3 coordinate transformation matrix (equation 6)

displacement vector

displacement vector at plane x = 0

displacement vector in x'y'z'-coordinate system

displacement vector at plane x' = 0 in x'y'zl-coordinate system

displacement vector for the second elastodynamic state in reciprocaltheoremdisplacement vector for the free-field motion

displacement vector for the free-field motion at plane x = 0

displacement vector for the free-field motion in x'y'z'-coordinatesystemdisplacement vector for the free-field motion at plane x' = 0 III

x' y'zl-coordinate systemdisplacement vector for the scattered field

displacement vector for the scattered field at plane x =0

{U}, Uxi , Uyi , Uzi vector containing nodal displacements at plane x = 0 and its com­ponents at the jth node

47

x,y,z

x',y',z'

x"' y., z"

'7,.

'7"

spatial coordinates

spatial coordinates in x' y' z'-coordinate system

spatial coordinates of source point

= (x,y,z), an arbitrary point

= (x', y', z'), an arbitrary point In x'y' z'-coordinate system

= (0, y, z), an arbitrary point at plane x = 0

= (O,y.,z.), a source point at plane x=o

number

number

numerical coefficient

canyon sm.'face

cross-section of the canyon at plane x = 0

half-space surface

cross-section of the half-space surface at plane x = 0

extent of the jth element

small contour of radius r. around the source point at plane x = 0

Kronecker delta function

natural coordinate

constant hysteretic damping factor for P wave

constant hysteretic damping factor for Rayleigh wave

constant l:ysteretic damping factor for shear wave

critical incidence angle for SV wave

azimuth incidence angle between the wave propagation plane andx-axis (Figure 2)angle between the normal to the incident wave front and the verticalaxis (Figure 2)shear wavelength

shear modulus for the half-space medium

Poisson's ratio for the half-space medium

48

V'

w

n

wavenumber in zl-direction

material density for the half-space medium

angles of tangent lines passing through a non-smooth boundarypoint on f c U f h

frequency in rad/sec.

non-dimensional frequency defined by n = wL/1rca

49

APPENDIX A: FREE-FIELD GROUND MOTIONS FOR SH, P, SVAND RAYLEIGH WAVES

By definition, the free-field motion is the displacement field that exists inside the

half-space in absence of the canyon. Except for incident Rayleigh surface wave, the

free-field motion consists of the incident wave and waves reflected by the half-space

surface. In the following expressions, the waves are assumed to travel parallel to

x' - Z' plane (see Figure 2). The vertical incidence angle is OIJ and the amplitudes of

the reflected P and S waves are Ap and A. respectively. In X' y'zl-coordinate system,

the free-field displacement fields can be expressed as [35]:

For Incident SH Wave

{uiJ(Xl)} = G) ~',I-'i""'+=""')i + (D .',I-" ••,.'-oo•••• ')i

For Incident P Wave

where

. c..s~n/, = -s~nOIJ

cp

A = sin(201J)sin(2/') - (Cp/C.)2cOS2(2/')p sin(201J)sin(2/') + (Cp/C.)2COS2(2/')

A _ 2(cp/c.)sin(201J)cos(2/')• - sin(201J)sin(2/') + (Cp/C.)2COS2(2/')

For Incident SV Wave

(A'1)

(A·2)

The free-field motions fc,r incident SV wave are given by two different expressions,

depending on whether the vertical incidence angle OIJ is smaller than the critical

incidence angle OIJ defined hy

50

where

• Cp • 0sznj = -szn vC$

A$ = sin(20v )sin(2j) + (Cp/C$)2COS2(20v)

sin(20v )sin(2j) - (Cp/C$)2COS2(Z8v)A

p= sin(20v )sin(2j) + (Cp/C$)2cos2(20v)

where

a = VPsin28 - k2$ v p

-2asin8vsin(20v) + k$ COS2 (20v)iAl = k$ COS2 (28v) - 2asinOvsin(20v)i

k$sin(48v )i

For Incident Rayleigh Wave

(, 2,)e-bL Z _ 1. (2 _ q. )e-bTZ

-. 2 C 'k '{u' (X')} = a • e-' .'"f f , 2,

ik [_h.e-bLZ + -1_(2 - .so. )e-bTZ ]r k~ 2bT C~

where

bL= kr (1 - C;r/2Cp

bT = k r (l- :~r/2

and Cr and k r are Rayleigh wave velocity and the corresponding wavenumber.

51

(A·3)

(A·4)

(A· 5)

APPENDIX B: FULL-SPACE GREEN'S FUNCTIONS

The full-space Green's functions can be found in Reference [38]. Let the source

and receiver points be XOII = (xoI, X0 2, X03) and x = (XI, X2, X3) respectively and n be

the normal to the surface at the receiver x with corresponding components denoted

by n = (nI,n2,n3) where, in order to make the formulation compact, the coordinate

notations have been changed from (x, y, z) to (Xl, X2, X3). The displacement and traction

components at x in the jth direction due to concentrated force at X08 in the kth

direction are given by

where j, k, 1= 1,2,3,

d2 = X2 - X0 2

and kp and k ll are P and S wavenumbers, respectively.

52

APPENDIX C: TRANSFORMED GREEN'S FUNCTIONS

Transformed Green's Displacement Functions

The transformed Green's displacement functions are defined (equation 18a) by

i,j=1,2,3 (C ·1)

where Gij(x,xos) is the ith displacement component at receiver point x = (X17X2,X3)

due to concentrated force at source point xos = (0, X 0 2, X 0 3) in the jth direction in full­

space. In order to make the formulations compact, the notation for the coordinates

has been changed from (x,y,z) to (XI,X2,X3). Correspondingly, the coordinates of the

source point become (0, X 0 2, X 0 3).

(C·2)i,j=1,2,3

From Reference [38], the Green's displacement functions, Gij, can be expressed

either in the form given in Appendix B or in the following more compact form

1 [1 [j2 (e- ik•

d- e- ikpd

) 600

0 ]G* - + t3 e-tk• d

ij - 41rp w2 [)Xj[)Xj d de;

where d is the distance between the source and receiver points

In order to obtain explicit expressions for Gij, it is necessary to use following

relation

(C·3)

where b is a complex constant, d is the distance between the source and receIver

projected to the plane Xl = 0:

and K 00 is modified Bessel function of the zeroth order.

Substituting (C· 2) into (C· 1) and utilizing (C· 3) lead to

Gll = 2~Jl [Ko(asd) (1 - ~;) + ~;Ko(apd)] (C ·4)

53

where

ap = Jk2 - k~

as = y'k2 - k;

j = 2,3

a,f3 = 2,3

(C·5)

(C·6)

and J1 is the shear modulus for the full-space medium. It should be pointed out that

k is a real variable while ap and as are complex numbers in general. When damping

is included, ap and as never vanish. Thus, Gij is always bounded for J =I o.

Transformed Green's Traction Functions

nThe transformed Green's traction functions Fij(xo, xos, k) are defined (equation

18b) by

i,j = 1,2,3 (C·7)

nwhere Fij(x,xos) is the ith traction component at the receiver point x = (X},X2,X3) due

nto a concentrated force at source point X08 = (0, X 0 2, X 0 3) in jth direction. Fij can be

nobtained by substituting the expressions of Ftj' which are given in Appendix B, into

(C . 7) and finding the integrations term by term. However, since finding derivativesn

is much easier than deriving integrations, it is more convenient to find Fij through

following ways:

(1) Assume a displacement field given by

G ( - - k) ikxlUij = ij x o, Xos, e (C·8)

(2) Find the corresponding stresses and tractions for this field and express the

latter as

(C·9)

n(3) Remove eikx1 from (C· 9) and one obtains Fjj

54

(C ·13)

nIt can be shown that Fij thus obtained are the same as what would be obtained

nby substituting Frj into equation (C· 7) and deriving the integrations term by term.

nThe final expressions for Fij are given by (j = 1,2,3)

F1j = Jl [n2 (88~:j + ikG2j ) + n3 (88~~j + ikG3j) ] (C· 10)

F2j = Jl{ 1~~v [ikVGlj + (1 - v) 8:X:j+ v 88~:j ] + n3 (88~:j + 88~:j) } (C ·11)

F3j = Jl{ 1~~v [ikVGlj + (1 - v) 8:X:j+ v 8:X:

j] + n2 (88~:j + 88~:j) } (C ·12)

where n2 and n3 are the components of outgoing normal to the surface, n = (0, n2, n3),

and v is the Poisson's ratio for the half-space medium.

Expression for Matrix [C]

Matrix [C] depends on the position of the source point; and, when the source

point is on the boundary, it is a function of the geometric shape of the system at

the source point. From equation (20), matrix [C] is defined by

[C(xos)] = lim f [F(xo,xos,k)fdrr.-o}r.

nSubstituting [F] into (C·13) and following similar procedures presented in Reference

[39], it can be found that

(C ·14)

where

1Cll = 21r (</>1 - </>2) (C ·15)

C22 = 41r(11_ v) [2(1- V)(</>1 - </>2) + %(Sin(2</>d - Sin(2</>2»] (C ·16)

C23 = C32 = (1 ) (sin2</>1 - sin2</>2) (C ·17)41r 1 - V

C33 = 41r(11_ v) [2(1- V)(</>1 - </>2) - %(sin(2</>d - Sin(2¢>2»] (C .18)

In the above equations, </>1 and </>2 are the angles of tangent lines passing through the

source point xos at the boundary. The detailed definition of </>1 and </>2 can be found

in Reference [39].

55

PART II

IMPEDANCE FUNCTIONS FOR THREE-DIMENSIONAL

FOUNDATIONS SUPPORTED ON AN INFINITELY­

LONG CANYON OF UNIFORM CROSS-SECTION IN A

HOMOGENEOUS HALF-SPACE

INTRODUCTION

Required in the substructure method for earthquake analysis of concrete dams is

the impedance matrix (or the frequency-dependent stiffness matrix) for the foundation

rock region, defined at the nodal points on the dam-foundation rock interface [1,2].

Computation of this foundation impedance matrix for analysis of arch dams requires

solution of a series of mixed boundary value problems governing the steady-state

response of the canyon cut in a three-dimensional, unbounded foundation rock region.

Because such analyses are extremely complicated, usually only the foundation flexibility

is considered in analysis of arch dams, i.e. the material and radiation damping as well

as inertial effects of the foundation rock are ignored [2-4]. The objective of this work

is to overcome these limitations with the long term goal of analyzing the earthquake

response of arch dams including dam-foundation rock interaction.

Computation of foundation impedance matrices has been the subject of many

investigations over the past two decades leading to a variety of analysis techniques.

Impedance matrices for two-dimensional embedded foundations have been determined

by the indirect boundary element method (BEM) with half-space Green's functions

[5], direct boundary element method with full-space Green's functions [6], and by a

hybrid method [7]. Procedures were developed to determine the impedance matrices

for three-dimensional surface-supported or embedded foundations by an integral equa­

tion technique with half-space Green's functions [8,9], nonsingular integral equation

approach [10-12], and a hybrid method combining indirect BEM with FEM using

half-space Green's functions [13].

Boundary methods utilizing half-space Green's functions have the advantage that

they are applicable to layered media and do not require discretization of the half-space

surface. However, because of the high cost in computing half-space Green's functions,

application of these methods has been restricted to either arbitrary-shape, surface­

supported foundations or axisymmetric embedded foundations. In contrast, full-space

59

Green's functions are simple to compute and have been used widely in the direct

BEM to determine foundation impedance matrices [14-19]. However, use of full-space

Green's functions seems computationally impractical for analysis of foundations In

layered media because of the requirement to discretize interfaces between layers.

In addition to various boundary methods using different Green's functions, other

techniques have been developed for evaluation of foundation impedance matrices.

These include the finite element method in the frequency domain with transmitting

boundaries [20-23], finite element method in the time domain [24], spherical wave

function expansion approach [25], infinite element approach [26,27] and various hybrid

methods [e.g. 28,29]. More extensive reviews of the various techniques developed to

determine foundation impedance matrices are available in [30,31].

The various techniques mentioned above have been applied to the analysis of

a wide variety of foundations either supported on the surface of a half-space or

embedded In a finite-sized excavation in the half-space. However, these techniques

appear to be either inapplicable or computationally impractical for the system of

Figure 1. It is an infinitely-long canyon of arbitrary but uniform cross-section cut

in a homogeneous viscoelastic half-space. In this work this system is analyzed to

determine the impedance matrix at the arch dam-foundation rock interface by the

direct boundary element method in conjunction with full-space Green's functions. The

uniform cross-section of the infinitely-long canyon, which seems to be a reasonable

idealization for practical analysis of arch dams, permits analytical integration along the

canyon axis leading to a series of two-dimensional boundary problems involving Fourier

transforms of the full-space Green's functions. The assumption of a homogeneous half­

space seems to be appropriate for arch dam sites where similar rock usually extends

to considerable depth.

PROBLEM STATEMENT

The system considered is an arch dam, idealized as a finite element system,

60

supported in an infinitely-long canyon of arbitrary but uniform cross-section cut in

a homogeneous half-space (Figure 1). The substructure method has proven to be

an effective approach to the earthquake analysis of dams including dam-foundation

rock interaction effects. These interaction effects introduce an impedance matrix, or

a dynamic (frequency-dependent) stiffness matrix, for the foundation rock region in

the equations of motion governing the steady-state response of the dam to harmonic

ground motion.

Our objective IS to determine the impedance matrix, [SAw)], where W IS the

frequency of the harmonic excitation, defined at the interface ri between the dam

and foundation rock. This impedance matrix, [Sf(w)], relates the interaction forces

{R(t)} at the interface ri to the corresponding displacements {ret)}, relative to the

earthquake induced ground displacements in the absence of the dam:

[SJ(w)]{r(w)} = {R(w)} (1)

where the overbar denotes Fourier transform of the time-functions. The square matrix

[Sf(w)] is of order equal to the number of degrees-of-freedom (DOF) in the finite

element idealization at the interface. The nth column of this matrix multiplied by

eiwt is the set of complex-valued forces required at the interface DOF to maintain

a unit harmonic displacement, eiwt , in the nth DOF with zero displacements in all

other DOF.

Evaluation of these forces requires solution of a senes of mixed boundary value

problems with displacements prescribed at the interface ri and tractions outside ri

- on the canyon as well as the half-space surface - are prescribed as zero. Instead

of directly solving this mixed boundary value problem, it is more convenient to solve

a stress boundary value problem in which non-zero tractions are specified at the

interface r i and the resulting displacements at r i are determined. Assembled from

these displacements, the dynamic flexibility influence matrix is inverted to determine

the matrix [Sfew )].

61

Fig.! Infinitely-long canyon of arbitrary but uniform cross-section; rc is thecross-section of the canyon at x = 0; r i is the interface between thedam and foundation rock.

62

The frequency-dependent impedance matrix [S/(w)] is function of the material

properties of the foundation rock and the geometry of both the canyon and the

dam-foundation rock interface. For the system considered, the material properties

of the half-space medium are characterized by the complex P and S wave velocities

Cp =cp(l + i77p)1/2 and Cs = cs(1 + i77s)1/2. The terms 77p and 77s represent the constant

hysteretic damping factors for P and S waves respectively; and cp and Cs denote the

P and S wave velocities if the medium were undamped. Similarly, the Rayleigh wave

velocity is given by Cr = cr(l + i77r)1/2 where Cs is the Rayleigh wave velocity without

damping and 77r is the corresponding hysteretic damping factor. Both Cr and 77r are

related to cp, cs , 77p and 778'

FOURIER REPRESENTATION OF TRACTIONS AND DISPLACEMENTS

Because the system geometry is uniform along the length of the canyon, it is

useful to express tractions and displacements through their Fourier integral represen­

tations. The tractions are prescribed over the dam-foundation rock interface, which is

discretized into two-dimensional curved boundary elements compatible with the finite

element discretization at the base of the dam. The prescribed traction on the jth

boundary element is {t(x)}jeiwt , where x = (x,y,z) is an arbitrary point within the

element, and the vector {t(x)}j contains the x, y and z components of tractions. The

traction vector is expressed through its Fourier integral representation:

(2)

where Xo = (0, y, z) and {i(io , k)}j is the Fourier transform of {t(i)}j with respect to x

(3)

The Fourier transform parameter k can be interpreted physically as a wavenumber

since {i( i o , k)} je- ikxeiwt represents tractions associated with a wave travelling along

the x-axis. Thus equation (2) suggests that the prescribed tractions can be considered

as a superposition over all wavenumbers of tractions associated with waves travelling

63

in opposite directions along the x-axis. In particular, a negative value of k indicates

waves travelling in the negative x-direction and a positive k represents waves moving

in the positive x-direction. The superposition of these waves, which travel at speed

c =w/lkl, results in the prescribed tractions in the form of a standing wave.

The prescribed tractions on the dam-foundation rock interface are next expressed

in terms of traction distributions over individual boundary elements. Along the line

parallel to the x-axis passing through xo , the traction distribution is given by the

superposition of tractions associated with each of the elements intersected by the line:

where

{i(xo , k)} = I)i(xo , k)}jj

(4)

(5)

The tractions are obviously zero at all locations on the canyon surface and half-space

surface that are outside the dam-foundation rock interface.

The steady-state displacements {u(x) }eiwt resulting from the prescribed tractions

with harmonic time-variation at frequency w can also be expressed as a superposition

of displacements associated with travelling waves. Thus, the vector {u( x)} containing

the three components of displacements at the canyon boundary r c, or the half-space

surface rh, resulting from the prescribed tractions (equation 4) is given by

(6)

where {u(xo , k)} is the Fourier transform of the displacement vector gIven by

{U(xo , k)} = 21 joo {u(x)}eikxdx7r -00

(7)

For any particular wavenumber k, {u(xo , k)}e- ikx represents a wave with displacement

pattern {u(xo , k)} travelling in the x-direction with speed c = w/lkl. As mentioned

before, a positive value of k indicates a wave travelling in the positive x-direction,

64

and vice-versa. In addition, the dependence of the displacements and tractions on

the frequency w is implied throughout.

We have now expressed the prescribed' tractions and unknown displacements

In terms of their Fourier integral representation which implies superposition over

all wavenumbers (equations 4 and 6). Considering one wavenumber at a time, the

prescribed boundary tractions are {t(xo,kne-ikxeiwt and the unknown displacements

are {u(xo,kne-ikxeiwt, where w is the excitation frequency and k is the wavenumber.

The problem to be solved for a particular wavenumber involves determining the

displacements {u( i o , kn at the curve f' c, where x = 0, due to the travelling wave

associated with tractions {t(io,kn obtained from equation (5) at x = o. The resulting

displacements for all wavenumbers are superposed to obtain the displacements at

any location i on the boundary (equation 6); in particular, the displacements at the

dam-foundation rock interface, f i , can be determined.

BOUNDARY INTEGRAL FORMULATION

The boundary integral equation for the system considered, consisting of an in­

finitely long canyon of uniform cross-section, can be obtained from the reciprocal

theorem and can be expressed as (Part I)

(8)

where {t(in is the prescribed traction at i on the boundary with its normal denoted byn

n, {u(xn is the resulting unknown displacement vector, and [F*(i, ios)] and [G*(i, ios)]

are 3 x 3 matrices of Green's functions for a full-space. The first, second and third

columns of these matrices correspond to the traction and displacement vectors at i

due to a unit point load applied in the x, y and z directions, respectively, at a source

point ios = (0, ys, zs) on f' c or f'h, the intersection line of the system boundary and

the plane x = o. The integration is over the entire system boundary, consisting of the

surface of the half-space and the boundary of the canyon, modified slightly to avoid

the singularity in the Green's functions at the point of load application (Figure 2).

65

Fig.2 Definition of boundaries and source point; rc is the cross-section ofthe canyon at x = 0, rh is the x = 0 line on the surface of the half­space, and r. is the small contour of radius T. around the sourcepoint.

66

Expressions for [F*(x, xos)] and [G*(x, xos)] are given III Appendix B of Part I. For

each source point, xos, equation (8) provides three scalar equations in the unknown

displacements {u( x)}.

Substituting equations (4) and (6) into equation (8) and interchanging the order

of integration lead to

=0

After analytically evaluating the integral along the x-axis, equation (9) becomes

(9)

where

(lla)

(llb)[F(xo,xos,k)] = i:[F*(fi,xos)]e-ikXdx

Expressions for the transformed Green's functions, [G] and [F], are given in Appendix

C of Part I.

As mentioned earlier, the displacements resulting from prescribed tractions

{f(xo,k)}e-ikxeiwt have the form {u(xo,k)}e-ikxeiwt, i.e., {f(xo,k)} and {u(xo,k)} corre­

spond to the same wave travelling at speed c = w/k, where k is the wavenumber.

Thus, equation (10) implies that the following equation must be satisfied for each k

The final step is to make the radius rs of the small arc f's around the source

point xos approach zero. Following the steps in [32], the final form of the boundary

integral equation (12) can be expressed as

67

(13)

where

[C(xos)] = lim f [F(xo, Xos, k)]Tdr (14)r.-O Jr.

The general expression for the 3 x 3 matrix [C(xos)] is given in Appendix C of Part

I. It is independent of the wavenumber k and depends only on the geometry of the

boundary at the source point X08" If the boundary is smooth, i.e., it has a unique

tangent, Cij(ios) = 6ij /2.

Equation (13) is the exact formulation for the problem, requiring solution of the

displacements {u(xo, k)} along f\ and f\, i.e. x = 0, due to tractions {f(io, k)}e- ikx

associated with a single wave travelling at speed c = w/k. In this equation, {f(xo,k)}

is known from equation (5), the transformed Green's function matrices [G] and [F]

are known from equation (11) and Appendix C of Part I, whereas the unknowns

are the displacements {u(xo,k)} at both the source point xos and the rest of the

boundary f\ u rh. Thus, by taking advantage of the uniform geometry of the system

along the x-direction, the displacements associated with a single travelling wave can

be determined by solving a "two-dimensional" boundary integral problem (equation

13) in which the remaining integration is only along the curve r c U rh •

Solution of an infinite number of such two-dimensional problems, each corre­

sponding to a particular wavenumber, would lead to the displacements {u(xo , k)} for

all wavenumbers, which are combined in accordance with equation (6) to determine

the displacements {u( i)} at any location on the boundary, and in particular at the

dam-foundation rock interface. In practice, of course, the infinite integration range

in equation (6) would be replaced by a finite range and the truncated integral would

be numerically evaluated from results at an appropriate set of wavenumbers.

BOUNDARY ELEMENT FORMULATION

Discretization

In its present form, equation (13) cannot be solved analytically for the displace-

68

ments {u( X0, k)}, even for the simplest canyon geometry. Therefore, the integration

domain feU f h is discretized into M one-dimensional line elements (Figure 3). Along

the half-space surface f h , the discretization obviously extends to only a finite dis­

tance L y , which should be large enough to provide accurate solutions. The number

of nodes in each element depends on the element type; in this study, two node el­

ements are used with linear interpolation functions given by N1(O = 0.5(1 - () and

N2(O = 0.5(1 + 0 where ( varies from -1 to 1. Thus the discretized system with M

elements has M n = M +1 nodes.

The variation of the unknown displacements {u(xo , k)} over the jth element on

feu f h can be expressed in terms of nodal displacements

where [N(O] is a 3 x 6 matrix consisting of interpolation functions

(15)

(

NI[N(O] = ~

N2 0o N 2

o 0(16)

and {Uh IS the vector of displacements at the two nodes - j and j + 1 - of the

element

As mentioned before, the dam-foundation rock interface f i IS discretized into

two-dimensional curved elements compatible with the finite element discretization at

the base of the dam. The total number of nodes on r i is denoted by Mi. On each

element, the prescribed tractions can be expressed in terms of nodal tractions; for

the jth element

(17)

where the matrix [h(x))j consists of two-dimensional interpolation functions and {T}j

is the vector of tractions at all the nodal points of the element. If the jth element

has M nodes, [h(x)]j is of size 3 x 3M and {TL is of dimension 3M. As a result, the

69

x

Fig.3 Boundary element discretization.

70

y

y

Fourier transforms of the prescribed tractions (equation 3) in the jth element on fj

can be represented by

(18)

where [h(xo,k)]j is the Fourier transform of the two-dimensional interpolation functions

with respect to x

(19)

Expressions of [h(xo,k)L for 4, 6 and 8 node elements are given in Appendix A.

Combining the contributions of all boundary elements on f j intersected by the

line x= Xo (equation 5), the Fourier transform of the traction distribution along this

line is given by

{t(xo,k)} = L [h(xo,k)]j{T}jj

a"'''emble

which, after assembly, can be expressed in matrix form as

(20)

(21)

In equation (21), {Tho contains the tractions at all the nodal points on the elements

intersected by the line x= xo •

Thus, after discretizing l'c ul'h into one-dimensional line elements and f j into two­

dimensional elements, the Fourier transforms of the unknown displacements, {u(xo , k)},

have been expressed in terms of nodal displacements (equation 15) and, {t(xo,k)}, the

Fourier transforms of the prescribed tractions, have been represented through nodal

tractions {T} (equation 21).

Boundary Element Equations

The next step is to replace the boundary integrations III equation (13) by the

summation of integrals over all the M elements

M

[C(xos)]{u(xos, k)} +L 1[F(Xo, Xos, k)]] {u(xo,k)}jdfj=l t j

71

M2 .

= L 1[G(Xo,xos,k)];{t(xo,k)}jdrj=M1 fj

(22)

where rj is the extent of the jth element. The summation on the right side only

covers the elements on the canyon boundary, rc, since the tractions are zero on the

half-space surface, rh. By expressing the displacements and the tractions in terms

of nodal displacements and tractions through equations (15) and (21) and changing

the integration variable to (, the contributions from the jth element in equation (22)

become

~ [G(xo, xos, k)]; {[(xo, k)}jdr = ~Ij11

[G((, Xos, k)]J[Ji((, k)]j{T},d( (23b)if j 2 -1

where Ij is the length of the jth element. Substituting these equalities, equation (22)

can be expressed in matrix form

(24)

Equation (24) represents three equations associated with each source point xos =

(0, Ys, zs) and each wavenumber k, where the 3 x 3 matrix [C(xos)] is given by equation

(14), the 3 x 1 vector {u(xos,k)} represents the unknown displacements at the source

point, {iT} is a 3Mn x 1 column vector of unknown nodal displacements (3 displacement

components at each of the Mn nodes on the boundary rcurh ), {T} is a 3Mi X 1 column

vector of nodal tractions (3 components at each of the M j nodes on r j ), [A(xos, k)]

is a 3 x 3Mn matrix assembled from the integrals of the products of the transformed

traction Green's functions and element interpolation functions

- 1[A(xos, k)] == 2 (25)

and [B(xos,k)] is a 3 x 3Mi matrix assembled from the integrals of the products of

the transformed displacement Green's functions and Fourier transform of the two-

72

dimensional interpolation functions (equation 21) ,

- _ 1 M 2 II _ T -[B(xos,k)] = - L lj [G((,xos,k)]j [H((,k)]jd(

2 ;=Ml -IG,uemole

(26)

Both [A(xos, k)] and [.8(xos, k)] are obtained by appropriately assembling the contri­

bution of the various elements, a process similar to the stiffness matrix assembly in

the direct stiffness method. The construction of both of these matrices for a small

system is illustrated in Figure 4.

Equation (24) is a discrete form of equation (13) obtained by introducing two

approximations: (1) interpolation of the unknown displacements over each element on

l"curh in terms of nodal displacements (in this particular formulation, the interpolation

functions were chosen to be linear); and (2) truncation of the integration on rh to a

finite distance.

Formation of Systenl Matrices

Associated with the source point xos = (0, Ys, zs), equation (24) represents 3 linear

algebraic equations in 3Mn unknown displacements. Clearly, a total of 3Mn equations

are needed to determine the unknown displacements. These equations can be obtained

by successively choosing the source point at each of the nodes of the discretized

boundary r c Urh. When the source point coincides with the ith node, equation (24)

becomes

(27)

By varying i = 1,2, ... , Mn and assembling the resulting equations, a system of 3Mn

linear equations can be obtained

[A(k)]{U} = [B(k)]{T}

where [A(k)] is a square matrix of order 3Mn , given by

(

[C(XI)] ) ([A(XI'k)])

[A(k)] = + .

[C(XMJ] [A(x~n,k)]

73

(28)

(29)

x

2

7 y

contribution ofelement 3 = [A(xo""k)h

s

contribution ofelement 3 = [B(xo""k)h

}{T}~==c;..;:.==c;..;:.="'=";;.I=.==...=..;;J

1•

Ufc[C(xos)]{u(xos, k)} +{

~~~~~

+

FigA Assembly of Matrices.

74

and [B(k)] is a 3Mn x 3Mi matrix given by

(

[B(.:l' k)J )

[B(k)] = _

[B(XMn ' k)J

(30)

The coefficient matrix [A(k)J is fully populated and unsymmetric, which is of typical

of direct boundary element method equations.

Evaluation of Displacements

Solution of the equation (28) leads to the displacements {tI} at the M n nodes on

the boundary f' c u f' h in terms of the nodal tractions {T} for a particular wavenumber

k

{tI} = [A(k)t1[B(k)]{T} (31)

Once the nodal displacements are determined, the displacement variation {u(x0' k)}

over each finite element on f' c can be obtained from equation (15).

IMPEDANCE MATRIX

After the displacements {u(xo , k)} at the y-z plane (x = 0) of the canyon boundary,

f' c, and the half-space surface, f'h, are obtained for various values of the wavenumber,

k, the displacements at any other location on these boundaries can be obtained from

equation (6). In particular, the displacements at the dam-foundation rock interface,

ri, can be determined. The indefinite integral of equation (6) is truncated at a finite

range (-K, K) and it is evaluated from the displacements obtained for a discrete set

of wavenumbers k q using an appropriate integration scheme:

{u(X)} = L Wq{u(xo , kq)}e-ikqxq

(32)

where Wq are weighting coefficients associated with the numerical integration scheme,

and kq is the qth sampling point in the wavenumber domain k. Guidelines for selecting

sampling points in the wavenumber domain will be presented later.

75

Thus, the vector of displacements at the jth node on fi, defined by the coordinates

Xj = (x j, Yj, Zj), due to the prescribed tractions is given by

{-} "W{-(- k)}-ikqxjr j = L.J q U Xoj, q eq

(33)

where i oj = (0, Yj, Zj), and {u(xoj,kq)} is obtained from the solution of equation (31)

for the nodal displacements and the displacement interpolation relation (15). If Xoj

falls in the mth element (Figure 3) in the plane x = 0 with (j as the corresponding

natural coordinate, {u(xoj,kq )} is given by (see equation 15)

(34)

where N 1((j) and N 2((j) are the linear interpolation functions given in the previous

section, and {U}m and {U}m+l are the 3 x 1 vectors of displacements at the mth and

(m+l)th nodes on fcufh associated with the qth sampling point k q • Since the nodal

displacements are linear combinations of the nodal tractions (equation 31), {u(xoj,kq )}

in equation (34) can also be expressed as linear combinations of {T}, leading to

(35)

where [E(xoj,kq)] is a 3x3Mi matrix relating the Fourier transform of the displacements

along the line x= Xoj to the nodal tractions. Thus from equations (33) and (35), the

vector of displacements at the jth node on f i becomes

where

[jj] = L Wq[E(xoj, kq)]e-ikqxjq

Combining equations (36) for each of the Mi nodes on fi leads to

{f} = [f]{T}

76

(36)

(37)

(38)

where {r} is a 3Mi x 1 vector of nodal displacements ( 3 components at each of the

Mi nodes) and [f] is a square matrix of order 3Mi given by

(

[id )[I] = _:

[lMi]

(39)

The matrix [f] in the above equation is the frequency-dependent flexibility influence

matrix which relates the displacements at the DOF on the dam-foundation rock

interface fi to the corresponding nodal tractions {T}.

As mentioned earlier, the impedance matrix [Sj(w)] for the foundation rock relates

nodal forces to nodal displacements at the discretized dam-foundation rock interface.

This matrix is obtained from equation (38) by transforming distributed tractions to

nodal forces:

{R} = [~]{T} (40)

where {R} is a 3Mi x 1 vector of nodal forces ( 3 components at each of the Mi nodes

on f i ), and [~] is a square matrix of order 3Mi assembled from the products of the

two-dimensional interpolation functions

[~] =Ii [h(x)f[h(x)]df

Combining equations (38) and (40) gives

{R(w)} = I~][I(w)]-l{r(w)}

(41)

(42)

wherein it is recognized that the nodal forces {R(w)}, the nodal displacements {few)}

and the flexibility influence matrix [few)] are frequency-dependent. Thus, comparing

equations (1) and (42) leads to the 3Mi x 3M; impedance matrix

[Sew)] = [~][f(w)]-l (43)

The impedance matrix determined from equation (43) will not be exactly sym­

metric due to the approximations inherent in the boundary element procedures; it is

77

nearly symmetric, however. By minimizing the 'differences in the unsymmetric off­

diagonal terms in a least square sense [16], a symmetric impedance matrix is given

by

(44)

For a rigid foundation on the surface of the canyon, the nodal displacements at

the foundation-rock interface can be expressed as

{r(w)} = [a]{ro(w)} (45)

where {ro(w)} is the 6 x 1 vector containing the six rigid-body DOFs of the foundation

and [a] is a 3Mi x 6 matrix relating the nodal displacements on r i with {ro(w)}. Thus

the 6 x 6 impedance matrix for the rigid foundation is given by

[Sr(w)] = [a]T[Sj(w)][a]

SUMMARY OF PROCEDURE

(46)

The boundary element procedure developed in the preceding sections to determine

the impedance matrix, or dynamic stiffness matrix, corresponding to an excitation

frequency w, for the foundation rock region, defined at the DOF of the dam-foundation

rock interface, may be summarized as follows:

1. Discretize ri, the dam-foundation rock interface, into two-dimensional surface ele­

ments, select interpolation functions [hex)], and determine their Fourier transforms

(equation 19). The expressions of these Fourier transforms for 4, 6 and 8 node

surface elements are available in Appendix A.

2. Discretize fe, the canyon boundary at x = 0, and f h, the half-space surface at x = 0,

into M line elements covering a range Lyon fh (Figure 3). If linear interpolation

functions are selected for each element, the discretized system contains M n = M +1

nodes.

3. Choose an appropriate integration scheme and its associated weighting coefficients

Wq and a discrete set of wavenumber sampling points kq (equation 32). Guidelines

78

for selection of kq and the truncated integration range (-K,K) are presented in

a later section. For each k = kq , repeat steps 4 to 7 to express the nodal

displacements {U} on f'cuf\ in terms of nodal tractions {T} on ri (equation 31).

4. Establish equation (27) which includes three linear algebraic equations associated

with Xi, the ith node on f' c U f'h' in 3Mn unknowns (3 displacement components

at each of the M n nodes) by the following steps:

(a) Compute the elements of the 3 x 3 matrix [C(Xi)] which depend on the geometry

of the boundary at the ith node; see Appendix C of Part 1. These elements

are given by Cjk(Xi) = bjk/2 if the boundary is smooth, i.e., it has a unique

tangent at Xi.

(b) Compute the 3 x 3Mn matrix [A(Xi, kq )] from equation (25) by assembling the

contributions of all the M elements, as described in Figure 4. Element contri­

butions are determined from the integrals of the transformed traction Green'sn

functions, [F], (equation 11b) and element interpolation functions. Analyticaln

expressions for [F] are available in Appendix C of Part 1.

(c) Compute the 3 x 3Mi matrix [B(Xi, kq )} from equation (26) by assembling the

contributions from all the elements on f'c, as described in Figure 4. The

line element contributions are determined from equation (26) wherein the

transformed displacement Green's functions, [GJ, are computed from equation

(l1a) and the assembled Fourier transforms of the two-dimensional interpolation

functions, [ill, from equation (21). Analytical expression for [G] is available

in Appendix C of Part I and the Fourier transforms of the t\Yo...dimensional

interpolation functions are given in Appendix A.

5. Repeat the computations summarized in step 4 for each of the nodes, i = 1,2, ... , Mn

to determine [C(Xi)], [A(Xi' kq )] and [B(Xi' kq )] for all the nodes.

6. Evaluate the square matrix [A(kq )] of order 3Mn and the rectangular matrix

[B(kq )] of size 3Mn x 3Mi for the boundary element system from the corresponding

79

individual nodal matrices (step 5) using equations (29) and (30).

7. Solve the system of 3Mn linear algebraic equations (28) to express the nodal

displacements {U} on f'c u f'h in terms of the nodal tractions {T} (equation 31).

8. For each sampling point k = kq , compute the 3 x 3Mi matrix [E(xoj, kq )] (equation

35) by interpolating the nodal displacements {U} determined in step 7.

9. Compute the 3 x 3Mi matrix flj(w)] from equation (37) wherein [E(xoj, kq )] for any

k q is known from step 8 and the wavenumber sampling points k q were selected

in step 3.

10. Repeat the computations summarized in step 9 for each of the nodes, j = 1,2, ..., Mi'

on fi and combine the resulting [jj(w)] according to equation (39) to obtain the

3Mi x 3Mi flexibility influence matrix [j(w)].

11. Compute the 3Mi x 3Mi transformation matrix [~] from equation (41) wherein

[h(x)] was determined in step 1.

12. Compute the square matrix [Sew)] of order 3Mi from equation (43) wherein [jew)]

and [~] are known from steps 10 and 11. The symmetric impedance matrix [Bj(w)]

is determined from equation (44).

13. Repeat the computations of steps 3 through 12 for each harmonic excitation

frequency w of interest to obtain the corresponding impedance matrix. The ex­

citation frequencies should be selected to cover the frequency range over which

the ground motion and dam response are significant.

NUMERICAL ASPECTS

Selection of Sampling Points in Wavenumber Domain

As mentioned in steps 3, 9 and 10 of the procedure summary, the computation to

determine the nodal displacements {U} is repeated for a discrete set of wavenumbers

kq distributed over a truncated range (-J(, J(). However, analyses are necessary only

for positive wavenumbers because they indirectly also provide the displacements as-

80

sociated with corresponding negative wavenumbers·(Appendix B). Having determined

the displacements {il} associated with various kq and {u(xo , kq )} from equation (34),

the displacements at the dam-foundation rock interface fi are determined by numeri­

cally evaluating the integral of equation (6). Since solution of the "two-dimensional"

problem to determine {u(xo,k q)} requires much more computational effort compared

to evaluation of the integral of equation (6), the integration scheme chosen should

require as few sampling points in k q as sufficient for required accuracy.

In order to properly select the discrete set of wavenumbers over the range (0, K),

the variation of {u(xo , k)} with k is examined next. This variation is plotted for the

system shown in Figure 5. It is an infinitely-long canyon with a semi-circular cross­

section of radius L and a dam-foundation rock interface of width b = 0.2L subjected

to uniform horizontal traction on the interface. The displacement-wavenumber plot

for point A is presented in Figure 5. For each wavenumber k, the displacement at

A is determined by implementing steps 4 to 7 of the procedure summary wherein

{T}T = (0,1,0, ... ,0,1,0) in equation (31). It is apparent that the displacements peak

sharply at the Rayleigh wavenumber: kr =l.07 for ao = 1 and kr = 4.29 for ao = 4

where a o = wLlcs is the normalized frequency. Thus finely-spaced wavenumber values

would be necessary in the vicinity of kr, whereas coarse spacing would suffice away

from k r •

Thus the wavenumber domain (0, K) is divided into several subdomains, each

with different step size, with the subdomain as well as step sizes depending on the

excitation frequency, and the size and discretization of the system. Typically the

four subdomains chosen were: (0, kr - L1k), (k r - L1k, kr + L1k), (k r + L1k, kr + L1k1 ) and

(k r +L1k1 ,IO where °< L1k < kr and L1k1 ~ 3L1k. The integration is truncated at kq = K

when the maximum amplitude of the displacements {D} is smaller than a specified

tolerance. The four-point Gauss integration scheme utilized is convenient to implement

and flexible to use. However, it has the disadvantage that, when accuracy needs to

be improved, the previous results cannot be reused and the entire computation has

81

yL L

_--- uniform horizontaltractions on r i

A

---1\

y

z

0.15

0.10

I- ,0 0 =.::L ~

0 II

tx II......... ,I1::3 ,I

,I

0.05 III,,

,J

0.00

I,IIIIIII

II 00

= 4II

",I,II I, I, I

I II I, I, I

I II I

, I, I

, I

,'~ ".. .. \ ,, ...

------10.1----------·10 I- - - - - . IO~I

o 2kr = 1.07

4 kr =4.29 6Wovenumber k

8 10

Fig.5 Variation of displacements {u(xo , k)} at A with wavenumber kj a o =wL/cs = 1 and 4, v = 1/3, .1]p = 1]s = 0.02.

82

to be repeated for a new set of wavenumbers.

Alternately, more complicated adaptive integration schemes may be employed to

permit automatic control of errors and reuse of results from previous calculation. Since

the actual integrand for the present problem is not one function but a matrix function

with many entries (equation 37), direct implementation of an adaptive procedure may

be inconvenient because large computer storage or input-output operation is needed

to save the previously computed results. Therefore, the discrete set of wavenumber

sampling points, kq , are determined from a simple test function, which has the main

features of the integrand.

From equation (31) and Appendix B, the nodal displacements on the boundary

I'cuI'h can be expressed as

(47)

where [Bc(k)] consists of the transformed displacement Green's functions, [G], and

[BH(k)] consists of the Fourier transforms of the two-dimensional interpolation func­

tions (equations 19 and 21), [H(xo,k)]. The simple test function may be obtained

by analyzing the main feature of each of the three matrices, i.e. [A(k)]-l, [Bc(k)]

and [BH(k)], in equation (47). First, the entries in [A(k)]-l have peaks of different

amplitudes near the Rayleigh wavenumber. Thus, the wavenumber dependence of

[A(k)]-l may be characterized by l/g(k) where g(k) is the Rayleigh function given by

[33]

(48)

Secondly, as k becomes greater than kr and further increases, the nodal displacements

{U} decrease which, as will be seen in the next subsection, is primarily controlled

by the asymptotic behavior of the modified Bessel functions Ko(asd) and K1(asd) in

[Bc(k)]. The slowest decay among all elements in [Bc(k)] with respect to k may be

controlled by KI(asdmin) where dmin is the shortest of all distances between a source

point, xos, and all the integration points on I'c U I'h used to evaluate equation (26).

83

Finally, the high frequency content in the variation of the nodal displacements {tT}

with respect to k is mostly due to the exponential terms in the matrix [BH(k)]. This

variation should be no faster than eiklxlmu where Ixl max is the maximum absolute value

of the x-coordinate for the nodes on the dam-foundation rock interface (Appendix A).

Combining the main features of the three matrices in equation (47), a test function

may be chosen as

4>(k) = J(1(;(:)in) eiklxlmu (49)

It should be pointed out that the 4>(k) of equation (49) does not represent any term

in the matrix [A(k)t 1 [Ba(k)][BH(k)] but is intended to represent the main variation

features of the matrix with respect to k. In particular, for a system with larger dmin

and Ixl max , equation (49) implies that finer sampling points over the wavenumber

range (0, J() should be selected, but the integration also can be truncated at a shorter

J(.

Based on this test function, any adaptive integration scheme, such as that based on

Chebychev approximation [34] or quartic polynomial interpolation [8], may be used

to determine the wavenumber sampling points, kq • In this investigation, a simple

adaptive integration scheme based on Simpson's rule was applied to the test function

to determine k q • In this scheme, the integration of the test function over a typical

interval, (k1 , k2 ), is first computed by

(50)

Then the integration is computed again by inserting two more sampling points midway

between the first three sampling points through

12 = (k2;;k1

) [4>(k1 ) +44>(3k1: k2

) + 24>( k1; k2

) + 44>( k1: 3k2

) +4>(k2 )] (51)

If the relative error between II and 12 is below a specified tolerance, the five sampling

points for the interval (kI, k2 ) are saved for later use. Otherwise, the process is

repeated for the two intervals (kI, (k1 + k2 )/2) and ((k1 + k2 )/2, k2) by adding more

84

sampling points. After the sampling points, k q , for the test function are determined,

the computation of the nodal displacements {U} is repeated for these kq and the

displacements at r i are determined by numerically evaluating the integral of equation

(6). The integration is truncated at kq = J( when both 1q'>(kq)1 and the maximum

amplitudes of the displacements {U} over the canyon surface are smaller than a

prescribed tolerance. Numerical results show that this adaptive scheme associated

with the test function of equation (49) works very well for a variety of systems and

frequencies.

Truncation and Discretization Errors

As mentioned earlier, the infinite boundary integral on the half-space surface in

equation (13) is replaced by an integral over a finite range, L y (equation 22). Thus,

the accuracy in the computation of matrix [A(Xi,kq )] summarized in step 4(b) of the

procedure summary is directly related to L y • It is found that the L y necessary to

achieve a desired accuracy mainly depends on the wavenumber k and the excita­

tion frequency w, and is related to the decay of the displacements {u(xo,k)} along

f\. Therefore, the variations of displacements on the half-space surface for various

wavenumber k q is examined. For this purpose, a rigid foundation of width b = 0.2L

on a semi-circular canyon of radius L (Figure 6a) is considered. The foundation in­

terface is discretized evenly into 12 four node elements along circumference. Since,

as mentioned in step 7 of the procedure summary, the displacements are expressed

III terms of the nodal tractions {T}, it is only necessary to show the variation of

displacements associated with some of the nodal tractions. The amplitudes of the

Fourier transforms of displacements along the canyon and the half-space boundaries,

f'c U f h , due to horizontal tractions associated with the first node on ri (Figure 6b)

are plotted for several wavenumbers in Figures 7 and 8. These displacem~nts are

obtained from the second column of the matrix [A(kq)]-l[B(kq)] (equation 31), i.e. by

specifying {T} = {O, 1,0, ... , O}T in the equation. The normalized frequencies for the

harmonic tractions are ao = wL/cs = 1 and 2 with corresponding Rayleigh wavenumbers

85

node 1L

x

Ly

z

f\

y

Fig.6a Dam-foundation interface on a semi-circular canyon.

y

Fig.6b Horizontal tractions associated with node 1.

86

0.000-10 -8 -6 -4

I I

-2 a 2Distance y/L

4\

6 8 10

Fig.7 Variation of the amplitudes of Fourier transforms of displacementswith distance; a o = 1.0, v = 1/3. Displacements are due to hori­zontal tractions associated with the first node for various values ofwavenumber, k.

87

0.000-10 -8 -6 -4 -2 0 2

Distance y/L4 6 8 10

Fig.8 Variation of the amplitudes of Fourier transforms of displacementswith distance; a o = 2.0, v = 1/3. Displacements are due to hori­zontal tractions associated with the first node for various values ofwavenumber, k.

88

kr =1.07 and kr =2.15.

The displacements decay slowly with distance for wavenumber k q close to the

Rayleigh wavenumber k r , not as slowly for k q < k r +0.5, and very fast for k q > k r +0.5

(Figures 7 and 8). An explanation for the dependence of the decay rate of {u(x,kq )}

with distance on k q can be obtained by examining the asymptotic expressions for the

transformed Green's functions where the decaying trend is dominated by the modified

Bessel functions /(o(asd) and /(1(asd) with as = Jk~ - k; and d being the distance

between source and receiver points (Appendix C of Part I). It should be noted that kr

is larger than, but very close to k s • Thus, when kq > kr +0 where 0 is a small positive

number depending on the size of the system, as is a complex number with positive

real part which increases with kq • For large arguments, the asymptotic expressions

for /(o(asd) and /(1(a sd) are given by

(52a)

(52b)

respectively. As a result, /(0 and /(1 decay exponentially with respect to d which

leads to rapid decay of {u(xo , k q )} along r h • When k q ~ kr + 0, the real part of as

is still positive, but it is very small ( for zero damping, e.g., Re(as) = 0 for ks < ks)

which makes the decay much slower.

Comparing Figures 7 and 8, it can be observed that the displacements decay

more rapidly along the half-space surface for high frequency excitation. Therefore,

for computational efficiency, the discretization range L y should vary with wavenumber

kq and frequency w.

For the rigid banded foundation shown in Figure 6, the impedance matrix with

respect to the mid-bottom point 0' of the foundation can be obtained from equation

89

(46) and expressed as

(53)

o LSxm 0LSyr 0 0000

L2Srr 0 0o L 2 S mm 0o 0 L 2 S tt

Sxx, Syy and Szz are the translational

Sxx 0 0o Syy 0o 0 Szz

o LSyr 0LSxm 0 0

o 0 0

where J.L is the shear modulus for the half-space,

impedance coefficients, Srr and Smm are the rocking impedance coefficients, Stt is

the torsional impedance coefficient, and Syr and Sxm are the coupling terms. All the

coefficients are in a non-dimensional form. The variation of impedance coefficients with

Ly is shown in Figures 9 to 12 where the smooth curves are obtained by connecting

the results at L y / >'8= 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0 through a cubic

spline approximation and the impedance coefficients are normalized by their "exact"

values which are obtained by using L y = 5>'8' In this example, different L y are selected

for k q 5 kr + 0.5 and k q > kr +0.5, and the L y appearing in the figures refers to the

discretization range used for k q :$ k r + 0.5. The relative errors decrease faster with

respect to L y for higher frequency and, for most coefficients, they become less than

two percent if Ly > 2>'8' For a o = 2, the relative errors for the imaginary parts of

the vertical impedance function Szz and the coupling term Syr are relatively large,

but they are within ±4%. Thus, for k q 5 k r + 6, L y should be at least 2>'8' Since

the shear wavelength for low frequency is larger than that for high frequency, larger

discretization range is required for low frequency in general. However, because k r

decreases with frequency and coarse discretization is adequate for low frequency, the

total computational cost due to large Ly for low frequency case does not necessarily

Increase.

For k q > k r +6, a shorter discretization range L y is adequate because of the rapid

exponential decay of the transformed Green's functions. In this investigation, L y for

kq > kr +6 is chosen according to

(54)

90

..........cJ 1.0.........

Q)

a::

0.9

1. 1--

~-- -~ 10 ---- ~- ....... -en . , -~-., -- ""---.........

Q) -a:: --

0.9 I I I I I I I

0 1 2 3 4

ly/As

Fig.9 Real parts of the impedance coefficients determined with various valuesof L y , defining the discretization range on the half-space surface. Theimpedance coefficients are normalized by their "exact" values. Resultsare presented for ao=2 and 4.

91

---

..-Vl~ 1.0'-'

Q)

a::

0.9

1.1

..-evi 1.0'-' -

Q)

a::

0.9o

I

------

I1

I I3

I

4

Fig.IO Real parts of the impedance coefficients determined with various valuesof L Y1 defining the discretization range on the half-space surface. Theimpedance coefficients are normalized by their "exact" values. Resultsare presented for ao=2 and 4.

92

-vr 1.0'-"

E

1.10 0 = 2

- - - - - - - - 0 0 = 4

0.9

1.1

-(/)~ 1.0'-"E

0.9

1.1

-elf 1.0'-"

E

0.9

1.1

-uf 1.0'-"

E

0.9

Fig.ll Imaginary parts of the impedance coefficients determined with variousvalues of L}/, defining the discretization range on the half-space surface.The impedance coefficients are normalized by their "exact" values.Results are presented for ao =2 and 4.

93

1.1

0.9

1.1

.......eJ 1.0

..-E

- 2= 4

0.9

1.1

.......(/)~ 1.0..-E

0.9

1.1

.......J 1.0..-E

0.94

~ 1.0..-E

Fig.12 Imaginary parts of the impedance coefficients determined with variousvalues of L]I' defining the discretization range on the half-space surface.The impedance coefficients are normalized by their "exact" values.Results are presented for ao=2 and 4.

94

which decreases with increasing k q • To ensure the accuracy of the results on fe,

however, L y should not decrease without limit. Since the errors in the computed

displacements near the end of the discretization on l'h are relatively large, it is

recommended that L y should be greater than the smaller one of As and the half

width of the canyon.

The size of elements on the half-space boundary f' h may vary depending on their

distances from the canyon. In this study, element sizes on l'h are chosen according to

an arithmetic series (I, 1+ ~I, 1+ 2~1, ... ) where the elements adjacent to the canyon

have the same size as their neighbors on f' c and the elements farthest away have

the size As /4. Furthermore, in order to avoid ill-conditioning or singularity in the

flexibility influence matrix [f] (equation 39), there are some restrictions on location of

nodes on the canyon boundary f' c relative to the locations of nodes on r i. Detailed

description and theoretical background for this restriction are given in Appendix C.

In summary, the following guidelines for selection of kq , J(, L y and the element

sizes are recommended:

1. The sampling points, k q , in the wavenumber domain should be chosen unevenly

with finely-spaced sampling points near the Rayleigh wavenumber and coarsely-spaced

sampling points elsewhere. Either piecewise Gauss integration scheme or an adaptive

integration procedure can be used to determine k q • The adaptive procedure may be

combined with a test function which characterizes the main features of the Fourier

transforms of the displacements and the integral in equation (6) may be truncated

when the magnitudes of the test function and the nodal displacements {iT} on canyon

boundary are smaller than a prescribed tolerance.

2. The choice of the integration range, L y , on the half-space surface should depend

on k q • For k q ~ k r + 6, where 6 is a small positive number depending on the' size of

the system, L y > 2As where As is the shear wavelength. For kq > kr +6, L y may be

chosen according to equation (54) with the restriction that L y is greater than the

95

smaller of two values: As and the half width of' the canyon.

3. The element size on both the dam-foundation rock interface, r i, and the

boundary f e should be kept smaller than As /5.

VERIFICATION AND COMPUTATIONAL COSTS

A computer program was developed to implement the step by step procedure

summarized earlier to determine the impedance matrix of the foundation rock region

shown in Figure 1 with reference to the degrees of freedom on the dam-foundation

rock interface rio Results for two systems are presented to demonstrate verification

of the present procedure.

The first system analyzed is a rigid square footing on the surface of a half-space for

which the impedances or compliances are available [10,18]. Figure 13 shows some of

the compliance functions determined by various methods for different non-dimensional

frequency ao = wb/cs where b is the half-width of the footing. The hysteretic damping

factors used in the present analysis are TIp = TIs = 0.001. In applying the present method,

the interface between the footing and the half-space is discretized into thirty six 8­

node, square elements, the line segment (-b,b), which is equivalent to fe, is discretized

into 12 elements, and the boundary fh on the half-space surface is discretized up

to L y = 2As for kq ~ k r +0.5 where As is the corresponding shear wavelength. The

agreement between various results shown in Figure 13 is satisfactory. The differences

may be, in part, due to the finite discretization range, L y , on the half-space surface,

which leads to a stiffer system in the present method, and the slight difference in

damping used in the various analyses.

The second system analyzed is an infinitely-long canyon of semi-circular cross­

section of radius L with a dam-foundation rock interface of width b = 0.2L (Figure

14). The impedance functions are determined for the interface ri assumed to be rigid

with six degrees of freedom. In implementing the present procedure, the interface is

discretized evenly into 12 elements along the circumferential direction and 2 elements

96

0.20

0.15

Wong & Luco [1 0]- - - - - - - - - Rizzo et.al. (18]

o Present Results

.01 0.10u

0.05

0.000.2

-----

41 2 3Normalized Frequency, 0 0

o0.0

.0

=!- 0.1J:

u

0.00.3

Real0.2..,

.0::t22u 0.1

Fig.13 Comparison of compliances obtained by present method with previousresults for vertical (Cvv), horizontal (CH H) and rocking (CM M )compliances of square footing supported on the surface of a viscoelastichalf-space (v = 1/3).

97

(a) Dam-foundation interface

Lx

O.1L

x

~~~~C L Ly y

(b) Plan view of a quarter of an illustrative (not actual) three-dimensionalboundary element discretization

Fig.14 Dam-foundation interface on a semi-circular canyon of radius L.

98

along the canyon axIS. The boundary l'c is discretized evenly into 12 line elements

and L y = 2As for kq < kr + 0.5. The impedances are presented in Figures 15 and

16 as a function of the normalized frequency ao = wL/cs . These impedance func­

tions result from cubic spline functions connecting computed impedance values for

ao = 0.2,0.5, 1.0, 1.5,2.0,2.5,3.0,3.5 and 4.0. Because previous results are apparently not

available for this system, the impedances were also obtained by analyzing the same

system by a three-dimensional boundary element method (equation 8). The interface

r j , a finite length Lx = L of the canyon, and the half-space surface up to L y = L

are discretized by two-dimensional boundary elements (Figure 14). The agreement is

satisfactory between the results obtained by the 3-D boundary element method and

by the procedure presented in this report.

Compared with the general 3-D boundary element approach, the present method

IS more accurate since the boundary integration in the direction of canyon axis IS

evaluated analytically. If highly-accurate results are desired, the present method IS

also more efficient. This is illustrated in Table 1 where the relative errors in the

impedance coefficients for the second problem described above are listed. The system

was analyzed by the 3-D boundary element method with L y = L and various values

of Lx, and by the present method with L y = L. Lacking an exact solution for this

problem, the comparison is performed against the solutions by the present method with

the discretization range L y = 2As • Clearly, the performance of the present method is

superior to that of the 3-D boundary element methods in accuracy and computational

time if results with errors less than 5% are desired.

The present method is especially efficient when the number of DOF along the

canyon axis is relatively large (Appendix B). Since the computer program implementing

the 3-D boundary method was not written with blockwise storage and, hence, is limited

by the computer storage capacity, relatively short discretization ranges L y and Lx

were used (Table 1). In general, however, such short discretization range are not

expected to lead to accurate results because the results are likely to oscillate with L y

99

6~--------------,

Real

_o-----o-----~----­---4

6-r---------------,5Real

,.. -----9 -----e----- -­-----

5

4

xx 3

(/)~3

(/)

2 ...... Imag./ao--0------0------0- _

2 Imag./ao- - E>- _~-----G--- __

o 2 3 4 o 1 2 3 4

(a) Translational Impedance So (b) Translational Impedance S»,

6-r---------------. 6--r---------------,

5 Real 5 Real

4D

_--..0. _--_--0----- 4 -----o-----O- ~----

Imag./ao--El_

- .. --'O"-----G-- _

432

Imag./co

_ -G-----e-----~----

o

2

.. 3en

4321o

2

NN 3

(/)

(c) Translational Impedance Su (d) Rocking Impedance S"

Fig.15 Comparison of foundation impedances obtained by the present method(dashed lines) with the 3-D boundary element method with Lx = L(symbols). The system analyzed is a rigid foundation of width 0.2Lon a semi-circular canyon of radius L (v = 1/3, Tip = Tis = 0.02).

100

4~-----------'----, 4-,.----------------,

.3 .3 Real

E 2E

(f) Real-----~-----o-----e-----

Imag.jao',---G-----e-----~----

:=2(J)

... "' .0. - - - - -, - -0 - - - - - ..,

Imag./ao

_-6 - - - - - 0_ - - - - -0- - - --

o 2 .3 4 o 2 3 4

(a) Rocking Impedance Smm (b) Torsional Impedance Sit

4--.----------------,

_---o-----~----­_ -G-

4

Imag.jao

2 3

Real

- - 9- - - - - -& - - - ­_ - - -0 - --

'--- G- -----0-'-- '"-1J"- _

o

o

-2

E -1"(J)

432

Real3~~2~

(J) ,

--1 Imag./ao1 ~ ..1----0 - - - - - 0- - - - - -<>- - - - -

o -L'...,.....,.-r-~-,--r-r-,-r--r--,--r_T""...,..__.....,.........,,..._1I I ,

o

(c) Coupling Impedance S1" (d) Coupling Impedance S"",

Fig.16 Comparison of foundation impedances obtained by the present method(dashed lines) with the 3-D boundary element method with Lx = L(symbols). The system analyzed is a rigid foundation of width O.2Lon a semi-circular canyon of radius L (v = 1/3, TJp = 7]8 = 0.02).

101

Table 1 Percentage Errors in Impedances and Computation Time for Two Methods

3-D Boundary Element Method with L", = Present MethodImpedances

0.0 0.25L 0.5L 0.75L 1.0L with L", = L

Re(S",,,,) 7.0 4.2 4.1 4.1 3.7 0.6

Im(S",,,,) 18.4 10.5 7.3 4.7 2.1 -1.3

Re(Syy) -7.9 3.6 4.0 2.6 1.3 0.2

Im(Syy) 15.3 8.7 1.6 -1.0 -1.5 2.6

Re(Szz) 1.8 3.2 2.2 1.9 1.9 2.0

Im(Szz) 10.4 6.3 5.0 5.3 5.4 -3.8

Re(Srr) -5.5 -1.1 1.1 2.6 2.9 0.8

ao = 2 Im(Srr) 7.8 8.2 6.9 4.5 1.1 3.0

Re(Smm) 13.9 7.8 4.2 2.7 2.1 0.5

Im(Smm) -3.1 -8.4 -6.3 -3.7 -2.2 -1.0

Re(Stt) -3.5 -6.6 -4.9 -1.8 0.8 0.0

Im(Stt) -9.8 -1.7 5.7 9.7 8.3 1.2

Re(Syr) -1.2 1.3 2.6 2.3 1.3 0.8

Im(Syr) 13.6 7.6 2.3 -1.1 -3.4 2.7

Re(S",m) 14.7 9.8 6.9 5.5 4.4 0.8

Im(S",m) 13.2 3.2 1.1 -0.02 -1.6 -1.7

Re(S",,,,) 0.02 -2.9 2.1 3.2 0.3 0.5

1m(S",,,,) -4.1 3.8 5.9 2.3 0.8 -0.1

Re(Syy) 4.8 4.2 2.8 3.8 4.3 0.2

Im(Syy) 4.4 -1.3 0.3 1.5 -0.4 1.3

Re(Szz) -1.3 -1.3 1.8 2.4 -0.8 1.0

Im(Szz) -1.3 2.3 2.9 0.6 0.04 0.1

Re(Srr) 9.9 3.2 0.5 1.5 5.1 -1.9

ao = 4 Im(Srr) 2.3 -3.6 -1.2 1.7 -0.8 1.7

Re(Smm) -9.4 -4.8 1.5 1.6 0.0 -0.1

Im(Smm) -0.6 7.6 7.6 4.1 4.5 -0.03

Re(Stt) 18.7 5.8 3.6 3.8 2.6 -0.3

Im(Stt) 3.6 -1.7 0.2 0.7 -2.0 0.6

Re(Syr) 14.5 2.0 0.9 2.3 3.8 -1.7

Im(Syr) 0.1 -0.8 1.6 3.8 0.7 2.8

Re(S",m) -8.0 -6.5 0.2 1.9 -0.3 0.5

Im(S",m) -7.2 4.2 7.0 4.0 3.5 -0.1

Compt.Time* (sec.) 3.1 7.9 15.6 26.8 41.4 24.0

* on Cray X-MP/14

102

and Lx (Table 1) until both discretization ranges extend two times the shear wave­

length.

The present method has the advantage that it requires much less storage compared

to the 3-D boundary method. Thus the accuracy of the present method can be

improved at extra computational cost by increasing the discretization range and

making the discretization finer. However, the accuracy of the 3-D boundary method

is likely to be limited by the available computer storage.

CONCLUSIONS

A direct boundary element procedure is presented to determine the impedance

matrix, i.e. the frequency-dependent stiffness matrix, associated with the nodal points

at the base of a structure supported on a canyon cut in a homogeneous viscoelastic

half-space. The canyon is infinitely-long and may be of arbitrary but uniform cross­

section. The uniform cross-section of the canyon permits analytical integration along

the canyon axis of the three-dimensional boundary integral equation. Thus the original

three-dimensional problem is reduced to an infinite series of two-dimensional boundary

problems, each of which corresponds to a particular wavenumber and involves Fourier

transforms of full-space Green's functions. Appropriate superposition of the solutions

of these two-dimensional boundary problems leads to a dynamic flexibility influence

matrix which is inverted to determine the impedance matrix.

A step-by-step summary of the procedure is presented and numerical aspects of

the method have been investigated. In particular, guidelines are presented for selection

of the sampling points in the wavenumber domain, the finite discretization range on

the half-space surface necessary to approximate the infinite range, and the element

size on the dam-foundation rock interface.

The accuracy of the procedure and implementing computer program has been

verified by comparison with previous results for a surface-supported, square foundation

and solutions for a foundation of finite-width on a infinitely-long canyon by a three-

103

dimensional boundary element method (BEM). Compared with the three-dimensional

BEM, the present method requires less computer storage and is more accurate and

efficient. Computation of the foundation impedance matrix by this method would

enable earthquake analysis of arch dams including dam-foundation rock interaction

effects:

104

REFERENCES·

1. G. Fenves and A.K. Chopra, "Earthquake Analysis of Concrete Gravity DamsIncluding Reservoir Bottom Absorption and Dam-Water-Foundation Rock Inter­actions," Earthquake Engineering and Structural Dynamics, 12, pp.663-680, 1984.

2. K.-L. Fok and A.K. Chopra, "Earthquake Analysis of Arch Dams Including Dam­Water Interaction, Reservoir Boundary Absorption and Foundation Flexibility,"Earthquake Engineering and Structural Dynamics, 14, pp.155-184, 1986.

3. P.S. Nowak, "Effect of Nonuniform Seismic In:2.ut on Arch Dams," Report No.EERL 88-03, California Institute of Technology, Pasadena, California, 1988.

4. Y. Ghanaat and R.W. Clough, "EADAP Enhanced Arch Dam Analysis Program,User's Manual," Report No. UCB/EERC-89/07, Earthquake Engineering ResearchCenter, University of California, Berkeley, Nov. 1989.

5. J.P. Wolf and G.R. Darbre, "Dynamic-stiffness Matrix of Soil by The BoundaryElement Method: Embedded Foundation," Earthquake Engineering and StructuralDynamics, 12, pp.401-416, 1984.

6. E. Alarcon, J. Dominguez and F.del Cano, "Dynamic Stiffness of Foundations,"New Developments in Boundary Element Methods, Proc. 2nd into seminar recent advancesboundary element methods, (ed. C.A. Brebbia) , pp.264-280, 1980.

7. T-J. Tzong and J. Penzien, "Hybrid Modeling of a Single-Layer Half-Space Systemin Soil-Structure Interaction," Earthquake Engineering and Structural Dynamics, 14,pp.517-530, 1986.

8. R.J. Apsel, "Dynamic Green's Functions for Layered Media And Applications toBoundary-value Problems," Ph.D. Dissertation, University of Califorma, San Diego,CA, 1979.

9. R.J. Apsel and J .E. Luco, "Impedance Functions for Foundations Embedded in aLayered Medium," Earthquake Engineering and Structural Dynamics, 15, pp.213-231,1987.

10. H.L. Wong and J .E. Luco, "Impedance Functions for Rectangular Foundations ona Uniform Viscoelastic Half-space," J. Geotechnical Engineering, ASCE, submittedfor publication.

11. J .E. Luco and H.L. Wong, "Response of Hemispherical Foundation Embeddedin Half-Space," J. Engineering Mechanics, ASCE, 112, No. 12, pp.1363-1374, Dec.1986.

12. H.L. Wong and J .E. Luco, "Tables of Impedance Functions for Square Foundationson Layerea Media," Soil Dynamics and Earthquake Engineering, 4, No.2, pp.64-81,1985.

13. C-H. Chen and J. Penzien, "Dynamic Modelling of Axisymmetric Foundations,"Earthquake Engineering and Structural Dynamics, 14), pp.823-840, 1986.

14. J. Dominguez, "Dynamic Stiffness of Rectangular Foundations," Research ReportR78-20, Dept. of Civil Engineering, MIT, Cambridge, MA, 1978.

15. J .M. Emperador and J. Dominguez, " Dynamic Response of Axisymmetric embed­ded Foundations," Earthquake Engineering and Structural Dynamics, 18, pp.1l05-1l17,1989.

16. A.P. Gaitanaros and D.L. Karabalis, "Dynamic Analysis of 3-D Flexible Embed­ded Foundations by a Frequency Domain BEM-FEM," Earthquake Engineering andStructural Dynamics, 16, pp.653-674, 1988.

105

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.34.

F.J. Rizzo, D.J. Shi2PY and M. Rezayat, "A Boundary Integr_al Equation Methodfor Time-harmonic Radiation and Scattering in an Elastic -Half-space," AdvancedTopics in Boundary Element Analysis, ASME, AMD, 72, pp.83-90, 1985.F.J. Rizzo, D.J. Shippy and M. Rezayat, "Boundary Integral Equation Analysisfor a Class of Earth-Structure Interaction Problems," Final Report, Dept. Eng.Mechanics, Univ. of Kentucky, Lexington, Kentucky, 1985.D.L. Karabalis and D.E. Beskos, "Dynamic Response of 3-D Embedded Founda­tions by The Boundary Element Method," Computer Methods in Applied Mechanicsand Engineering, 56, pp.91-119, 1986.M.J. Kaldjian, "Torsional Stiffness of Embedded Footings," Soil Mech. Found. Div.,ASCE, 95, No.SM1, pp.969-980, 1969.C.M. Urlich and R.L. Kuhlemeyer, "Coupled Rocking and Lateral Vibrations ofEmbedded Footings," Can. Geotech., 10, 2, pp.145-160, 1973.E. Kausel, J.M. Roesset and G. Waas, "Dynamic Analysis of Footings on LayeredMedia," J. Engineering Mechanics, ASCE, 101, No.EM5, pp.679-693, 1975.E. Kausel and J.M. Roesset, "DJnamic Stiffness of Circular Foundations," J.Engineering Mechanics, ASCE, 101, No.EM6, pp.771-785, 1975.S.M. DaY7,.T"Finite Element Analysis of Seismic Scattering Problems," Ph.D. Dis­sertation, university of California, San Diego, CA, 1977.V.W-S. Lee, "Investigation of Three-Dimensional Soil-Structure Interaction," Re­port CE 79-11, Dept. of Civil Engineering, University of Southern California, LosAngeles, CA, 1979.Y.K. Chow and I.M. Smith, "Infinite Elements for Dynamic Foundation Analysis,"Numerical Methods in Geomechanics, (Ed. Z. Eisenstein), 1, pp.15-22, 1982.

F. Medina and J. Penzien, "Infinite Elements for Elastodynamics," EarthquakeEngineering and Structural Dynamics, 10, pp.699-709, 1982.S. Gupta J. Penzien, T.W. Lin and C.S. Yeh, "Three-Dimensional Hybrid Mod­elling of Soil-Structure Interaction," Earthquake Engineering and Structural Dynamics,10, pp.69-87, 1982.A. Mita and W. Takanashi, "Dynamic Soil-Structure Interaction Analysis byHybrid Method," Boundary Elements, C.A. Brebbia, T. Futagami and M. Tanaka,Eas., pp.785-794, 1983.J.E. Luco, "Linear Soil-structure Interaction: A Review," Earthquake Ground MotionAnd Its Effects on Structures, AMD-Vol. 53, S.K. Datta (Ed.). American Societyof Mechanical Engineers, New York, pp.41-57, 1982.D.E. Beskos, "Boundary Element Methods in Dynamic Analysis," Applied MechanicsReviews, 40, 1, pp.1-23, 1987.F. Hartmann, "Computing the C-matrix in non-smooth boundary points," New De­velopments in Boundary Element Methods, (Ed. by C.A.Brebbia), CML Publications,pp.367-379, 1980.K.F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, 1975.Pei-cheng Xu and A.K. Mal, "An Adaptive Integration Scheme for IrregularlyOscillatory Functions," Wave Motion, 7, pp.235-243, 1985.

106

[A]

b

[B)

[13]

c

[C(X)]

[I]

[1]

NOTATION.

non-dimensional frequency defined by ao = wL/c.

= Jp -k;=Jk2

- k;square coefficient matrix for the boundary element system (equation29)

3 x 3Mn matrix assembled from the integrals of the products of[Ff and [N] associated with a single source point (equation 25)width of a foundation or half-width of a footing

matrix assembled from [B] (equation 30)

3 x 3Mi matrix assembled from the integrals of the products of [G]Tand [R] associated with a single source point (equation 26)matrices for the decomposed expression of matrix [B]

=w/kP wave velocity including damping

P wave velocity without damping

Rayleigh wave velocity including damping

Rayleigh wave velocity without damping

shear wave velocity including damping

shear wave velocity without damping

3 x 3 matrix related to the geometric shape of the boundary at x(equation 14)

= J(y - y.)2 + (z - z.)2, distance between source and receiver pointsprojected on plane x = 0

3 x 3 diagonal matrix defined by [b] = diag(l, -1, -1)

diagonal matrices consisting of [b) (Appendix B)

3 x 3Mi ma.trix relating the Fourier transform of the displacementsto the nodal tractions at r i (equation 35)flexibility influence matrix

3 x 3Mi matrix through which nodal displacements on r i are ex­pressed in terms of {T}

107

[F]

[F-]

g(k)

[G]

[G-]

rhO]

rhO]

k

K

I

L

M

M

transformed Green's traction functions (equation llb)

full-space Green's traction functions

= (2Jc2 - k;)2 - 4k2(k2 - k;)l/2(Jc2 - k;)l/2, Rayleigh function

transformed Green's displacement functions (equation lla)

full-space Green's displacement functions

matrix of interpolation functions for two-dimensional elements on

r.Fourier transform of the two-dimensional interpolation functions

rhO]matrices enlarged from matrices {~} and ['11]-1 (Appendix A)

matrix assembled from [hOl

=Aintegration of the test function t/>(k) over an interval

wavenumber in x-direction

qth sampling point in the k wavenumber domain

Rayleigh wavenumber

shear wavenumber

values in the wavenumber domain

integration limit in the k wavenumber domain at which the infiniteintegral of equation (6) is truncatedmodified Bessel functions of second kind of order zero and orderone

length of an element on f'c U f'h

half width of a two dimensional canyon

discretization range along the direction of canyon axIS for threedimensional boundary element approachdiscretizat on range on the half-space surface

number of one-dimensional elements on the boundaries f c U f'h

number of nodes on an element

number of Gauss integration points on l'c

number of nodes on the dam-foundation rock interface r i

108

{P} l P", , Py , P",

{r}

{il}

[8]

number of nodes on the boundaries f c U f h

element number for the first and the last one-dimensional elementson boundaryfc

outgoing normal to the boundary and its components

shape function matrix for the one-dimensional element on f c ufh ;

jth shape functionvector and components of concentrated forces associated with thesecond elastodynamic statevector of nodal displacements at the dam-foundation rock interface

6 x 1 vector containing the six degrees of freedom for a rigid foun­dationradius of the circular contour around a source point in plane x = 0

vector of nodal forces at the dam-foundation rock interface r i

unsymmetric impedance matrix resulted from boundary elementapproximationsimpedance matrix

impedance matrix for a rigid foundation

Stt, Smm, S",m, Syr non-dimensional impedance coefficients for rigid foundation

t

{t};t""ty,tz

{l};f"" fy, t",

{T}

{T",}, {Ty }, {T",}

{u}

{iL}

x,y,z

time

traction vector and its components

vector of Fourier transform of the surface tractions and its com­ponentsvector of nodal tractions

vector containing the x-, y-, and z-components of nodal tractions

displacement vector

Fourier transform of the displacement vector {u}

vector containing nodal displacements at plane x = 0 and its com­ponents at the jth nodeweighting coefficients associated with the integration scheme usedin k wavenumber domainspatial coordinates

109

x',y',z'

x" y" z,

[aJ

'1r

.A,

1I

{¢}

spatial coordinates in x'y'z'-coordinate system

coordinates of a point inside an element on r i

spatial coordinates of source point

= (x, y, z), an arbitrary point

a point in x'y'z'-coordinate system

= (0, y, z), an arbitrary point at plane x = 0

= (0, y" z,), source point at plane x = 0

3M. x 6 matrix relating nodal displacements on r i with the SIX

rigid-body degrees of freedom of a rigid foundationcanyon surface

cross-section of the canyon at plane x = 0

half-space surface

cross-section of the half-space surface at plane x = 0

dam-foundation rock interface

extent of the jth element

small contour of radius r, around the source point at plane x = 0

positive number

Kronecker delta function

natural coordinate

constant hysteretic damping factor for P wave

constant hysteretic damping factor for Rayleigh wave

constant hysteretic damping factor for shear wave

shear wavelength

shear modulus for the half-space medium

Poisson's ratio for the half-space IT_edium

test function defined in equation (49)

matrix that transforms distributing tractions into concentratednodal forces (equation 41)

row matrix defined in equation (A.3)

110

{,,&}j "&i[\l1]

w

Fourier transform of {1/J} and its jth component

square matrix defined in equation (A.4)

frequency in rad/sec.

111

APPENDIX A: FOURIER TRANSFORMS OF TWO-DIMENSIONAL

SHAPE FUNCTIONS

First, consider a two-dimensional curved 8 node element at the dam-foundation

rock interface fi (Figure A.l) where any point, X, on the element can be uniquely

represented by two local coordinates x' = (x', yl) (or (x', Zl) if the element is on a

vertical plane). The local and the global coordinate systems are related through

x = Xc +x'

y = Yc + y'

where (xc, Yc) are the global coordinates of a point inside the element.

(A.la)

(A.lb)

As described in the main text, the prescribed traction, {t(x)}, contains the x, y

and z components. For the purpose of illustration, consider the X component, tx(x),

first. At any point x' on the element, the traction tx(x' ) can be expressed in terms

of tractions at all the nodal points, x~, i=I,... ,8:

(A.2)

(A.3)

(AA)

where

({¢(~d})

[\II] = .

{¢(xs)}

and {Tx} = {Txt, ... , Txs}T is a vector containing the x-components of the nodal tractions

t --I '-1 8a Xi' Z- , ••• , •

Utilizing equations (A.l) and (A.2), the Fourier transform of tx(x) can be found

to be

112

x

x'(~,y)

IIIII

:(~,Yc y'I

(X01

,Y)II

o y

Fig.A.l Top VIew of an arbitrary element on dam-foundation interface.

113

(A.5)

where X o1 and X o2 are the global x-coordinates of the points (Figure A.I) at the element

boundary intersected by a line parallel to the x-axis passing through x' = (0, v'). In

equation (A.5), {~} is a 1 x 8 matrix containing the Fourier transforms of {'I/7(i')}

where

- 1 i Ok ok'1/71 = (e l X 0 2 _ el X 0 1 )

21r k

.1. 1[ i( ) 1] ikx 1[i ( ) 1] ikx0/2 = 211' -k X o2 - Xc + k2 e 02 + 21r k X o1 - Xc - k2 e 01

~3 = ~lY'

- 1 ikx [1 2 2 2i]'1/74 = 21r e 02 ik (X o2 - xc) + k 2 (X o2 - xc) + k3

1 ikx 0 1 [ 1 ( )2 2 ( ) 2i]- 211' e ik Xol - Xc + k2 Xol - Xc + k3

~5 = ~2Y'

~6 = ~ly'2

~7 = ~4Y'

~8 = ~2y'2

(A.6)

Similarly, the Fourier transforms of the prescribed tractions in the y and z direc­

tions can be expressed as

114

(A.7)

(A.8)

where {Ty} and {Tz } are the y and z components of the nodal tractions, respectively.

Combining equations (A.5), (A.7) and (A.8) leads to the expression for the 3 x 24

matrix [h(xo,k)] (equation 19) for the 8 node element

(A.9)

where [hI] is a 3 x 24 matrix enlarged from matrix {~} by replacing each entry in

{~} by a 3 x 3 diagonal matrix diag[1/Ji' ~i, 1/Ji], i = 1, ... ,8; and [h2 ] is a 24 x 24 matrix

enlarged from matrix ["iI!]-I in a similar manner.

For 4 and 6 node elements, the solution procedure is the same except that {¢(x')}

in equation (A.3) is replaced by {'ljJ(i')} = {l,x',y',x'y'} for 4 node element, and by

{'ljJ(x')} = {1,x',y',x12 ,x'y',y'2} for 6 node element.

115

APPENDIX B: RELATION BETWEEN {Uh AND {U}-k

As mentioned in the procedure summary, the computations to determine the

nodal displacements {U} are repeated for a discrete set of wavenumbers over a trun­

cated wavenumber range (-K, K). Due to a special property of the integrands with

respect to k, however, {U} for negative wavenumbers can be obtained indirectly from

{U} corresponding to positive wavenumbers. From Appendix C in Part I, the trans-n

formed Green's functions [G] and [F] corresponding to negative wavenumber -k can

be expressed as

where [D] is a 3 x 3 diagonal matrix given by

[D] = diag(l, -1, -1)

(B.1a)

(B.1b)

(B.2)

Thus, the coefficient matrix [A(k)] in equation (28), which consists of the in-n

tegrals of the products of the transformed traction Green's functions [F] and the

one-dimensional interpolation functions, has the property that

[A( -k)] = [D][A(k)][D]

where [D] is a diagonal matrix of order 3Mn

[D] = diag(l, -1, -1, 1, -1, -1, ..., 1, -1, -1)

From equation (B.3), it is apparent that

[A( _k)]-l = [D][A(k)r1[D]

since [D]-l = [D].

(B.3)

(BA)

(B.5)

The matrix [B(k)] in equation (28) consists of the integrals of the products of [G],

the transformed displacement Green's functions, and [H(io , k)], the Fourier transforms

116

of the two-dimensional interpolation functions on the dam-foundation rock interface r i

(equations 26 and 30). These integrals are evaluated numerically over each elements

on i:'c through Gauss integration scheme. Since [H(xo,k)] does not depend on the

source point xos , matrix [B(k)] can be decomposed as

[B(k)] = [Ba(k)][BH(k)] (B.6)

where matrix [Ba(k)] consists of the transformed displacement Green's functions asso­

ciated with different source points xos and receiver points xo , i.e. the Gauss integration

points; and matrix [BH (k)] consists of [H(x0' k)] for various receiver points. The sizes

of [Ba(k)] and [BH(k)] depend on the total number of Gauss integration points on

f'c' If the total number of Gauss integration points on f'c is Mg , the sizes for [Ba(k)]

and [BH(k)] are 3Mn x 3Mg and 3Mg x 3Mi , respectively. Due to the relation given

in equation (RIa) and the fact that [H(xo, -k)] is complex conjugate of [H(xo,k)],

matrix [B(k)] apparently has the property that

[B( -k)] = [D][Ba(k)][D1]conjg([BH(k))) (B.7)

where [D1] is a square diagonal matrix of dimension 3Mg having similar form as [D]

in equation (BA).

Thus, from equation (31), the nodal displacements on the boundary :rc u :rhare

given by

(B.8)

for positive wavenumber k and

(B.9)

for negative wavenumber -k. Once [A(k)t1[Ba(k)] is found, both {Uh and {U}-k

can be determined easily.

In general, it is more computationally efficient to use equations (B.8) and (B.9),

instead of equation (31), to solve for the nodal displacements {U} on the boundaries

117

f'cUf'h for both k and -k, especially when the total number of nodes on r i is greater

than the total number of Gauss integration points on f'c, i.e. Mi > Mg. Since the

computation of [A(k)]-l[BG(k)] does not depend on the nodal distribution at r i along

the x-direction, the computational cost for the present method does not increase

much when the number of nodes at r i increases along the x-direction. Therefore, the

present method is especially efficient when the number of DOF along the canyon axis

is relatively large.

118

APPENDIX C: RESTRICTIONS ON ·MESH LAYOUT ON f e

Two types of discretization are involved in the present method. One is the

discretization of l'e U f h , the system cross-section at x = 0, into one-dimensional line

elements and the other is the discretization of the dam-foundation rock interface, ri,

into two-dimensional surface elements. In order to avoid singularity problem in the

flexibility matrix, there is a certain restriction on the relative relation between the

two types of discretization.

In order to illustrate the restriction, consider a simple example shown in Figure C.l

where a banded foundation of constant width is resting on a semi-circular canyon. In

this example, l'e is discretized evenly into 4 line elements with the left most node being

the mth node and r i is discretized evenly into 8 elements along the circumferential

direction. From equation (33), the nodal displacements, {r}, on C is determined by

superposing the Fourier transforms {u(ioj,kq)} at fe' where {u(ioj,kq )} is determined

from their nodal values through equation (34). Thus, for this particular example, the

displacements at the first three nodes on ri are given by

{rh = I:Wq[Nl((d{Uq}m + N2((1){Uq}m+t]e-ikqXlq

{r}z = L Wq[N1((2){Uq}m + N2((2){Uq}m+I]e-ikqX2q

{rh = L Wq[NI((3){Uq}m + N2((3){Uq}m+1]e-ikqX3q

(C.l)

(C.2)

(C.3)

where (1, (2 and (3 are the corresponding natural coordinates for the three nodes.

Since Xl = X2 = X3, it is clear that these nodal displacements can be expressed in

matrix form as

(CA)

Because the second matrix on the right hand side of equation (C.4) is a linear

combination of the nodal tractions, {T} (see equation 35), equation (C.4) implies that

119

L

x

y

L

i\01.__---_...£----

y

z

Fig.C.1 lllustration of inappropriate mesh that will lead to singular flexibilityinfluence matrix.

120

the first three rows in the flexibility influence matrix [I] (equations 38 and 39) is a

linear combination of two rows. Thus the rank of the first three rows is less than 3

which means that the resulting flexibility influence matrix is singular.

From this example, it is clear that, in order to avoid singularity or ill-condition

problem in the flexibility influence matrix, the size of the elements on f e should be

smaller than or about the same as that of elements on rio More specifically, situations

similar to the example just illustrated, where three or more nodes having the same

x-coordinates on r i fall in one element on fe, should be avoided.

121

EARTHQUAKE ENGINEERING RESEARCH CENTER REPORT SERIES

EERC reports are available from the National Information Service for Earthquake Engineering(NISEE) and from the National Technical InformationService(NTIS). Numbers in parentheses are Accession Numbers assigned by the National Technical Information Service; these are followed by a price code.Contact NTIS, 5285 Port Royal Road, Springfield Virginia, 22161 for more information. Reports without Accession Numbers were not available from NTISat the time of printing. For a current complete list of EERC reports (from EERC 67-1) and availablity information, please contact University of California,EERC, NISEE, 130 I South 46th Street; Richmond, California 94804.

UCB/EERC-81101 "Control of Seismic Response of Piping Systems and Other Structures by Base Isolation; by Kelly, J.M., January 1981, (PB81 200735)A05.

UCB/EERC-81/02 "OPTNSR· An Interactive Software System for Optimal Design of Statically and Dynamically Loaded Structures with NonlinearResponse; by Bhatti, M.A., Ciampi, V. and Pister, K.S., January 1981, (PB81 218 851)A09.

UCB/EERC-81103 "Analysis of Local Variations in Free Field Seismic Ground Motions; by Chen, J.-C., Lysmer, J. and Seed, H.B., January 1981, (AD­A099508)AI3.

UCB/EERC-81/04 "Inelastic Structural Modeling of Braced Offshore Platforms for Seismic Loading; by zayas, V.A., Shing, P.·S.B., Mahin, SA andPopov, E.P., January 1981,lNEL4, (PB82 138 777)A07.

UCB/EERC-81105 "Dynamic Response of Light Equipment in Structures; by Der Kiureghian, A., Sackman, J.L. and Nour-Dmid, B., April 1981, (PB81218497)A04.

UCB/EERC-81/06 "Preliminary Experimental Investigation of a Broad Base Liquid Storage Tank; by Bouwkamp, J.G., Kollegger, J.P. and Stephen, R.M.,May 1981, (PB82 140 385)A03.

UCB/EERC-81/07 "The Seismic Resistant Design of Reinforced Concrete Coupled Structural Walls: by Aktan, A.E. and Benero, V.V., June 1981, (PB82113 358)AIl.

UCB/EERC-81/08 "Unassigned; by Unassigned, 1981.

UCB/EERC-81/09 "Experimental Behavior of a Spatial Piping System with Steel Energy Absorbers Subjected to a Simulated Differential Seismic Input," byStiemer, S.F., Godden, W.G. and Kelly, J.M., July 1981, (PB82 201 898)A04.

UCB/EERC-81/l0 "Evaluation of Seismic Design Provisions for Masonry in the United States; by Sveinsson, B.I., Mayes, R.L. and McNiven, H.D.,August 1981, (PB82 166 075)A08.

UCB/EERC-81/l1 "Two-Dimensional Hybrid Modelling of Soil-Structure Interaction," by Tzong, T.-J., Gupta, S. and Penzien, J., August 1981, (PB82 142118)A04.

UCB/EERC-81/l2 -Studies on Effects of Infills in Seismic Resistant RIC Construction: by Brokken, S. and Bertero, V.V., October 1981, (PB82 166190)A09.

UCB/EERC-81/13 "Linear Models to Predict the Nonlinear Seismic Behavior of a One-Story Steel Frame; by Valdimarsson, H., Shah, A.H. andMcNiven, H.D., September 1981, (PB82 138 793)A07.

UCB/EERC-81/l4 "TLUSH: A Computer Program for the Three-Dimensional Dynamic Analysis of Earth Dams: by Kagawa, T., Mejia, L.H., Seed, H.B.and Lysmer, J., September 1981, (PB82 139 940)A06. .

UCB/EERC-81/15 -Three Dimensional Dynamic Response Analysis of Earth Dams: by Mejia, L.H. and Seed, H.B., September 1981, (PB82 137 274)A 12.

UCB/EERC-81/16 "Experimental Study of Lead and Elastomeric Dampers for Base Isolation Systems: by Kelly, J.M. and Hodder, S.B., October 1981,(PB82 166 182)A05.

UCB/EERC-81/17 "The Influence of Base Isolation on the Seismic Response of Light Secondary Equipment: by Kelly, J.M., April 1981, (PB82 255266)A04.

UCB/EERC-81118 -Studies on Evaluation of Shaking Table Response Analysis Procedures: by Blondet, J. M., November 1981, (PB82 197 278)AI0.

UCB/EERC-81119 -DELIGHT.STRUCT: A Computer-Aided Design Environment for Structural Engineering," by Balling, R.J., Pister, K.S. and Polak, E.,December 1981, (PB82 218496)A07.

UCB/EERC-81/20 "Optimal Design of Seismic-Resistant Planar Steel Frames: by Balling, R.J.. Ciampi, V. and Pister, K.S., December 1981, (PB82 220I79)A07.

UCB/EERC-82101 "Dynamic Behavior of Ground for Seismic Analysis of Lifeline Systems: by Sato, T. and Der Kiureghian, A., January 1982, (PB82 218926)A05.

UCB/EERC-82102 "Shaking Table Tests of a Tubular Steel Frame Model; by Ghanaat, Y. and Cough, R.W., January 1982, (PB82 220 161)A07.

UCB/EERC-82/03 -Behavior of a Piping System under Seismic Excitation: Experimental Investigations of a Spatial Piping System supported by Mechani­cal Shock Arrestors," by Schneider,S., Lee, H.-M. and Godden, W. G., May 1982, (PB83 172 544)A09.

UCB/EERC-82/04 "New Approaches for the Dynamic Analysis of Large Structural Systems: by Wilson, E.L., June 1982, (PB83 148 080)AOS.

UCB/EERC-82105 "Model Study of Effects of Damage on the Vibration Properties of Steel Offshore Platforms: by Shahrivar, F. and Bouwkamp, J.G.,June 1982, (PB83 148 742)AIO.

UCB/EERC-82106 "States of the Art and Pratice in the Optimum Seismic Design and Analytical Response Prediction of RIC Frame Wall Structures; byAktan, A.E. and Benero, V.V., July 1982, (PB83 147 736)A05.

UCB/EERC-82107 "Further Study of the Earthquake Response of a Broad Cylindrical Liquid-Storage Tank Model: by Manos, G.c. and Gough; R.W.,July 1982. (PB83 147 744)A I!.

UCB/EERC-82108 "An Evaluation of the Design and Analytical Seismic Response of a Seven Story Reinforced Concrete Frame," by Charney, F.A. andBertero. V.V., July 1982, (pB83 157 628)A09.

UCB/EERC-82/09 "Fluid-Structure Interactions: Added Mass Computations for Incompressible Fluid: by Kuo, J.S.-H., August 1982, (PB83 156 281)A07.

UCB/EERC-82/l0 'Joint-Dpening Nonlinear Mechanism: Interface Smeared Crack Model: by Kuo, J.S.-H., August 1982, (PB83 149 195)A05.

123

UCB/EERC-82/11 "Dynamic Response Analysis of Techi Dam," by Clough, R.W., Stephen, RM. and Kuo, J.S.-H., August 1982, (PB83 147 496)A06.

UCB/EERC-82/12 "Prediction of the Seismic Response of RIC Frame-Coupled Wall Structures," by Aktan, A.E., Bertero, V.V. and Piazzo, M., August1982, (PB83 149 203)A09.

UCB/EERC-82/13 -Preliminary Report on the Smart I Strong Motion Array in Taiwan," by Bolt, B.A., Loh, C.H., Penzien, J. and Tsai, Y.B., August1982, (PB83 159 400)AIO.

UCB/EERC"82114 "Seismic Behavior of an EccentricalIy X-Braced Steel Structure," by Yang, M.S., September 1982, (PB83 260 778)AI2.

UCB/EERC-82/15 "The Performance of Stairways in Earthquakes: by Roha, C., Axley, J.W. and Bertero, V.V., September 1982, (PB83 157 693)A07.

UCB/EERC-82/16 "The Behavior of Submerged Multiple Bodies in Earthquakes: by Liao, W.-G., September 1982, (PB83 158 709)A07.

UCB/EERC-82/l7 "Effects of Concrete Types and Loading Conditions on Local Bond"Slip Relationships: by CowelI, A.D., Popov, E.P. and Bertero, V.V.,September 1982, (PB83 153 577)A04.

UCB/EERC.82/18 "Mechanical Behavior of Shear WalI Vertical Boundary Members: An Experimental Investigation: by Wagner, M.T. and Bertero, V.V.,October 1982, (PB83 159 764)A05.

UCB/EERC.82/19 "Experimental Studies of Multi-support Seismic Loading on Piping Systems: by Kelly, J.M. and Cowell, A.D., November 1982, (PB90262684)A07.

UCB/EERC-82/20 "Generalized Plastic Hinge Concepts for 3D Beam·Column Elements," by Chen, P. F.-S. and Powell, G.H., November 1982, (PB83 247981)AI3.

UCB/EERC-82/21 "ANSR-II: General Computer Program for Nonlinear Structural Analysis," by Oughourlian, C.V. and Powell, G.H., November 1982,(PB83 251 330)AI2.

UCB/EERC·82122 "Solution Strategies for StaticalIy Loaded Nonlinear Structures: by Simons, J.W. and Powell, G.H., November 1982, (PB83 197970)A06.

UCB/EERC-82/23 "Analytical Model of Deformed Bar Anchorages under Generalized Excitations: by Ciampi, V., Eligehausen, R., Bertero, V.V. andPopov, E.P., November 1982, (PB83 169 532)A06.

UCB/EERC"82/24 "A Mathematical Model for the Response of Masonry Walls to Dynamic Excitations: by Sucuoglu, H., Mengi, Y. and McNiven, RD.,November 1982, (PB83 169 OII)A07.

UCB/EERC-82/25 "Earthquake Response Considerations of Broad Liquid Storage Tanks: by Cambra, F.J., November 1982, (PB83 251 215)A09.

UCB/EERC·82/26 "Computational Models for Cyclic Plasticity, Rate Dependence and Creep," by Mosaddad, B. and Powell, G.H., November 1982, (PB83245 829)A08.

UCB/EERC·82127 "Inelastic Analysis of Piping and Tubular Structures: by Mahasuverachai, M. and Powell, G.H., November 1982, (PB83 249 987)A07.

UCB/EERC-83/0l "The Economic Feasibility of Seismic Rehabilitation of Buildings by Base Isolation: by Kelly, J.M., January 1983, (PB83 197 988)A05.

UCB/EERC-83/02 "Seismic Moment Connections for Moment-Resisting Steel Frames.: by Popov, E.P., January 1983, (PB83 195 412)A04.

UCB/EERC-83/03 -Design of Links and Beam-ta-Column Connections for Eccentrically Braced Steel Frames: by Popov, E.P. and Malley, J.O., January1983, (PB83 194 811 )A04.

UCB/EERC·83/04 "Numerical Techniques for the Evaluation of Soil·Structure Interaction Effects in the Time Domain: by Bayo, E. and Wilson, E.L.,February 1983, (PB83 245 605)A09.

UCB/EERC-83/05 "A Transducer for Measuring the Internal Forces in the Columns of a Frame-Wall Reinforced Concr::te Structure: by Sause, R. andBertero, V.V., May 1983, (PB84 119 494)A06.

UCB/EERC·83/06 "Dynamic Interactions Between Floating Ice and Offshore Structures," by Croteau, P., May 1983, (PB84 119 486)A16.

UCB/EERC·83/07 'Dynamic Analysis of Multiply Tuned and Arbitrarily Supported Secondary Systems," by Igusa, T. and Der Kiureghian, A., July 1983,(PB84 118 272)AII.

UCB/EERC·83/08 "A Laboratory Study of Submerged Multi·body Systems in Earthquakes: by Ansari, G.R., June 1983, (PB83 261 842)AI7.

UCB/EERC-83/09 "Effects of Transient Foundation Uplift on Earthquake Response of Structures: by Vim, C.-S. and Chopra, A.K., June 1983, (PB83 261396)A07.

UCB/EERC-83/10 'Optimal Design of Friction·Braced Frames under Seismic Loading," by Austin, M.A. and Pister, K.S., June 1983, (PB84 119 288)A06.

UCB/EERC·83/11 'Shaking Table Study of Single-Story Masonry Houses: Dynamic Performance under Three Component Seismic Input and Recommen­dations: by Manos, G.c., Clough, R.W. and Mayes, RL., July 1983, (UCB/EERC·83/11)A08.

UCB/EERC·83/12 -Experimental Error Propagation in Pseudodynamic Testing," by Shiing, P.B. and Mahin, S.A., June 1983, (PB84 119 270)A09.

UCB/EERC·83/13 "Experimental and Analytical Predictions of the Mechanical Characteristics of a 1/5"scale Model of a 7"story RIC Frame-Wall BuildingStructure: by Aktan, A.E., Bertero, V.V., Chowdhury, A.A. and Nagashima, T., June 1983, (PB84 119 213)A07.

UCB/EERC"83/14 "Shaking Table Tests of Large-Panel Precast Concrete Building System Assemblages: by Oliva, M.G. and Oough, RW., June 1983,(PB86 110 210/AS)AII.

UCB/EERC·83/15 "Seismic Behavior of Active Beam Links in Eccentrically Braced Frames: by Hjelmstad, K.D. and Popov, E.P., July 1983, (PB84 119676)A09.

UCB/EERC"83/16 "System Identification of Structures with Joint Rotation," by Dimsdale, J.S., July 1983, (PB84 192 21O)A06.

UCB/EERC"83/17 "Construction of Inelastic Response Spectra for Single.Degree-of-Freedom Systems: by Mahin, S. and Lin, J., June 1983, (PB84 208834)A05.

UCBIEERC·83/18 "Interactive Computer Analysis Methods for Predicting the Inelastic Cyclic Behaviour of Structural Sections: by Kaba, S. and Mahin,S., July 1983, (PB84 192 012)A06.

UCB/EERC·83/19 "Effects of Bond Deterioration on Hysteretic Behavior of Reinforced Concrete Joints: by Filippou, F.C., Popov, E.P. and Bertero, V.V.,August 1983, (PB84 192 020)AIO.

124

UCB/EERC-83/20 "Correlation of Analytical and Experimental Responses of Large-Panel Precast Building Systems; by Oliva, M.G., Clough, R.W., Vel­kov, M. and Gavrilovic, P., May 1988, (PB90 262 692)A06.

UCB/EERC-83121 'Mechanical Characteristics of Materials Used in a 115 Scale Model of.a 7-Story Reinforced Concrete Test Structure; by Bertero, V.V.,Aktan, A.E., Harris, H.G. and Chowdhury, A.A., October 1983, (PB84 193 697)A05.

UCB/EERC·83122 'Hybrid Modelling of Soil-Structure Interaction in Layered Media; by Tzong, T.-J. and Penzien, J., October 1983, (PB84 192 178)A08.

UCB/EERC-83123 'Local Bond Stress-Slip Relationships of Deformed Bars under Generalized Excitations,' by Eligehausen, R., Popov, E.P. and Bertero,V.V., October 1983, (PB84 192 848)A09.

UCB/EERC·83124 'Design Considerations for Shear Links in Eccentrically Braced Frames; by Malley, J.O. and Popov, E.P., November 1983, (PB84 192I86)A07.

UCB/EERC-84/01 'Pseudodynamic Test Method for Seismic Performance Evaluation: Theory and Implementation; by Shing, P.-S.B. and Mahin, S.A.,January 1984, (PB84 190 644)A08.

UCB/EERC-84/02 'Dynamic Response Behavior of Kiang Hong Dian Dam; by Clough, R.W., Chang, K.-T., Chen, H.-Q. and Stephen, R.M., April 1984,(PB84 209 402)A08.

UCB/EERC-84/03 'Refined Modelling of Reinforced Concrete Columns for Seismic Analysis; by Kaba, SA and Mahin, SA, April 1984, (PB84 234384)A06.

UCB/EERC-84/04 •A New Floor Response Spectrum Method for Seismic Analysis of Multiply Supported Secondary Systems; by Asfura, A. and DerKiureghian, A., June 1984, (PB84 239 417)A06.

UCB/EERC-84/05 'Earthquake Simulation Tests and Associated Studies of a 1/5th-scale Model of a 7-Story RIC Frame-Wall Test Structure,' by Bertero,V.V., Aktan, A.E., Charney, EA. and Sause, R., June 1984, (PB84 239 409)A09.

UCB/EERC-84/06 "RIC Structural Walls: Seismic Design for Shear; by Aktan, A.E. and Bertero, V.V., 1984.

UCB/EERC-84/07 'Behavior of Interior and Exterior Flat-Plate Connections subjected to Inelastic Load Reversals,' by Zee, H.L. and Moehle, J.P., August1984, (PB86 117 629/AS)A07.

UCB/EERC-84/08 'Experimental Study of the Seismic Behavior of a TWO-Story Flat-Plate Structure,' by Moehle, J.P. and Diebold, J.W., August 1984,(PB86 122 553/AS)AI2.

UCB/EERC-84/09 'Phenomenological Modeling of Steel Braces under Cyclic Loading; by Ikeda, K., Mahin, S.A. and Dermitzakis, S.N., May 1984, (PB86132 198/AS)A08.

UCB/EERC-84/10 'EarthQuake Analysis and Response of Concrete Gravity Dams; by Fenves, G. and Chopra, A.K., August 1984, (PB85 193902lAS)A11.

UCB/EERC-841l1 -EAGD-84: A Computer Program for Earthquake Analysis of Concrete Gravity Dams; by Fenves, G. and Chopra, A.K., August 1984,(PB85 193 613/AS)A05.

UCB/EERC-84/12 'A Refined Physical Theory Model for Predicting the Seismic Behavior of Braced Steel Frames; by Ikeda, K. and Mahin, S.A., July1984, (PB85 191 450/AS)A09.

UCB/EERC-84/13 -Earthquake Engineering Research at Berkeley - 1984; by, August 1984, (PB85 197 341/AS)AIO.

UCB/EERC-84/14 'Moduli and Damping Factors for Dynamic Analyses of Cohesionless Soils; by Seed, H.B., Wong, R.T., Idriss, LM. and Tokimatsu, K.,September 1984, (PB85 191 468/AS)A04.

UCB/EERC-84/15 -The Influence of SPT Procedures in Soil Liquefaction Resistance Evaluations; by Seed, H.B., Tokimatsu, K., Harder, L.E and Chung,R.M., October 1984, (PB85 191 7321AS)A04.

UCB/EERC-841l6 'Simplified Procedures for the Evaluation of Settlements in Sands Due to Earthquake Shaking; by Tokimatsu, K. and Seed, H.B.,October 1984, (PB85 197 887/AS)A03.

UCB/EERC-84/17 'Evaluation of Energy Absorption Characteristics of Highway Bridges Under Seismic Conditions - Volume I (PB90 262 627)A16 andVolume II (Appendices) (PB90 262 635)AI3; by Imbsen, R.A. and Penzien, J., September 1986.

UCB/EERC-84/18 'Structure-Foundation Interactions under Dynamic Loads; by Liu, W.D. and Penzien, J., November 1984, (PB87 124 889/AS)AII.

UCB/EERC-84/19 'Seismic Modelling of Deep Foundations; by Chen. e.-H. and Penzien, J., November 1984, (PB87 124 798/AS)A07.

UCB/EERC-84/20 'Dynamic Response Behavior of Quan Shui Dam; by Clough, R.W., Chang, K.-T., Chen, H.-Q., Stephen, R.M., Ghanaat, Y. and Qi,J.-H., November 1984, (PB86 115177/AS)A07.

UCB/EERC-85/0 I 'Simplified Methods of Analysis for Earthquake Resistant Design of Buildings; by Cruz, E.F. and Chopra, A.K., February 1985, (PB86112299/AS)AI2.

UCB/EERC-85/02 'Estimation of Seismic Wave Coherency and Rupture Velocity using the SMART I Strong-Motion Array Recordings; by Abrahamson,N.A., March 1985, (PB86 214 343)A07.

UCB/EERC-85/03 -Dynamic Properties of a Thirty Story Condominium Tower Building; by Stephen, R.M., Wilson, E.L. and Stander, N., April 1985,(PB86 118965/AS)A06.

UCB/EERC-85/04 'Development of Substructuring Techniques for On·Line Computer Controlled Seismic Performance Testing," by Dermitzakis, S. andMahin, S., February 1985, (PB86 13294I1AS)A08.

UCB/EERC-85/05 •A Simple Model for Reinforcing Bar Anchorages under Cyclic Excitations,' by Filippou, F.e., March 1985, (PB86 112 919/AS)A05.

UCB/EERC-85/06 'Racking Behavior of Wood-framed Gypsum Panels under Dynamic Load; by Oliva, M.G., June 1985, (PB90 262 643)A04.

UCB/EERC-85/07 'Earthquake Analysis and Response of Concrete Arch Dams; by Fok, K.-L. and Chopra, A.K., June 1985, (PB86 1396721AS)AIO.

UCB/EERC-85/08 "Effect of Inelastic Behavior on the Analysis and Design of Earthquake Resistant Structures; by Lin, J.P. and Mahin, S.A., June 1985,(PB86 135340/AS)A08.

UCB/EERC-85/09 'Earthquake Simulator Testing of a Base-Isolated Bridge Deck; by Kelly, J.M.• Buckle, LG. and Tsai, H.-C., January 1986. (PB87 1241521AS)A06.

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UCB/EERC-85/10 "Simplified Analysis for Earthquake Resistant Design of Concrete Gravity Dams: by Fenves, G. and Chopra, A.K., June 1986, (PB87124 160/AS)A08.

UCB/EERC-85/11 "Dynamic Interaction Effects in Arch Dams: by Clough, R.W., Chang, K.-T., Chen, H.-Q. and Ghanaat, Y., October 1985, (PB86135027/AS)A05.

UCB/EERC-85/12 "Dynamic Response of Long Valley Dam in the Mammoth Lake Earthquake Series of May 25-27, 1980: by Lai, S. and Seed, H.B.,November 1985, (PB86 142304/AS)A05.

UCB/EERC-85113 -A Methodology for Computer-Aided Design of Earthquake-Resistant Steel Structures: by Austin, M.A., Pister, K.S. and Mahin, S.A.,December 1985, (PB86 159480/AS)AlO .

UCB/EERC"85114 "Response of Tension-Leg Platforms to Vertical Seismic Excitations: by Liou, G.-S., Penzien, J. and Yeung, R.W., December 1985,(PB87 124 8711AS)A08.

UCB/EERC-85/1S -Cyclic Loading Tests of Masonry Single Piers: Volume 4 - Additional Tests with Height to Width Ratio of 1: by Sveinsson, B.,McNiven, H.D. and Sucuoglu, H., December 1985.

UCB/EERC-85/16 -An Experimental Program for Studying the Dynamic Response of a Steel Frame with a Variety of Inlill Partitions: by Yanev, B. andMcNiven, H.D., December 1985, (PB90 262 676)AOS.

UCB/EERC-86/01 -A Study of Seismically Resistant Eccentrically Braced Steel Frame Systems: by Kasai, K. and Popov, E.P., January 1986, (PB87 124178/AS)AI4.

UCB/EERC-86/02 -Design Problems in Soil Liquefaction: by Seed, H.B., February 1986, (PB87 124 186/AS)A03.

UCB/EERC-86/03 -Implications of Recent Earthquakes and Research on Earthquake-Resistant Design and Construction of Buildings: by Bertero, V.V.,March 1986, (PB87 124 194/AS)A05.

UCB/EERC-86/04 -The Use of Load Dependent Vectors for Dynamic and Earthquake Analyses: by Leger, P., Wilson, E.L. and Clough, R.W., March1986, (PB87 124 202/AS)AI2.

UCB/EERC-86/0S -Two Beam-To-Column Web Connections: by Tsai, K.-c. and Popov, E.P., April 1986, (PB87 124 301lAS)A04.

UCB/EERC-86/06 -Determination of Penetration Resistance for Coarse-Grained Soils using the Becker Hammer Drill: by Harder, L.F. and Seed, H.B.,May 1986, (PB87 124 210/AS)A07.

UCB/EERC-86/07

UCB/EERC-87/03

UCB/EERC-86/10

UCB/EERC-86111

UCB/EERC-86/08

UCB/EERC-86/09

"A Mathematical Model for Predicting the Nonlinear Response of Unreinforced Masonry Walls to In-Plane Earthquake Excitations,' byMengi, Y. and McNiven, H.D., May 1986, (PB87 124 780/AS)A06.

-The 19 September 1985 Mexico Earthquake: Building Behavior: by Bertero, V.V., July 1986.

-EACD-3D: A Computer Program for Three-Dimensional Earthquake Analysis of Concrete Dams: by Fok, K.-L., Hall, J.F. andChopra, A.K., July 1986, (PB87 124 228/AS)A08.

-Earthquake Simulation Tests and Associated Studies of a 0.3-Scale Model of a Six-Story Concentrically Braced Steel Structure: byUang, e,-M. and Bertero, V.V., December 1986, (PB87 163 S64/AS)AI7.

-Mechanical Characteristics of Base Isolation Bearings for a Bridge Deck Model Test: by Kelly, J.M., Buckle, I.G. and Koh, c.-G.,November 1987, (PB90 262 668)A04.

"Effects of Axial Load on Elastomeric Isolation Bearings: by Koh, C.-G. and Kelly, J.M., November 1987.

"The FPS Earthquake Resisting System: Experimental Report," by Zayas, V.A., Low, S.S. and Mahin, S.A., June 1987.

-Earthquake Simulator Tests and Associated Studies of a O.3-Scale Model of a Six-Story Eccentrically Braced Steel Structure,- by Whit­taker, A., Uang, C.-M. and Bertero, V.V., July 1987.

"A Displacement Control and Uplift Restraint Device for Base-Isolated Structures: by Kelly, J.M., Griffith, M.C. and Aiken, I.D., April1987.

UCB/EERC-87/04 "Earthquake Simulator Testing of a Combined Sliding Bearing and Rubber Bearing Isolation System: by Kelly, J.M. and Chalhoub,M.S., 1987.

UCB/EERC-86/12

VCB/EERC-87/01

UCB/EERC-87/02

UCB/EERC-87/0S "Three-Dimensional Inelastic Analysis of Reinforced Concrete Frame-Wall Structures: by Moazzami, S. and Benero, V.V., May 1987.

UCB/EERC-87/06 "Experiments on Eccentrically Braced Frames with Composite Floors: by Rides, J. and Popov, E., June 1987.

UCB/EERC"87/07 "Dynamic Analysis of Seismically Resistant Eccentrically Braced Frames: by Ricles, J. and Popov, E., June 1987.

UCB/EERC-87/08 "Undrained Cyclic Triaxial Testing of Gravels-The Effect of Membrane Compliance," by Evans, M.D. and Seed, H.B., July 1987.

UCB/EERC-87/09 "Hybrid Solution Techniques for Generalized Pseudo-Dynamic Testing," by Thewalt, C. and Mahin, S.A., July 1987.

UCB/EERC-87/10 "Ultimate Behavior of Butt Welded Splices in Heavy Rolled Steel Sections: by Bruneau, M., Mahin, S.A. and Popov, E.P., September1987.

UCB/EERC-87/I1 "Residual Strength of Sand from Dam Failures in the Chilean Earthquake of March 3, 1985: by De Alba, P., Seed, H.B., Retamal, E.and Seed, R.B., September 1987.

UCB/EERC-87112 "Inelastic Seismic Response of Structures with Mass or Stiffness Eccentricities in Plan: by Bruneau, M. and Mahin, S.A., September1987, (PB90 262 650)AI4.

UCB/EERC-87113 "CSTRUCT: An Interactive Computer Environment for the Design and Analysis of Earthquake Resistant Steel Structures," by Austin,M.A., Mahin, S.A. and Pister, K.S., September 1987.

UCB/EERC-87/14 -Experimental Study of Reinforced Concrete Columns Subjected to Multi-Axial Loading: by Low, S.S. and Moehle, J.P., September1987.

UCB/EERC-87/IS "Relationships between Soil Conditions and Earthquake Ground Motions in Mexico City in the Earthquake of Sept. 19, 1985: by Seed,H.B., Romo, M.P., Sun, J., Jaime, A. and Lysmer, J., October 1987.

UCB/EERC-87/16 "Experimental Study of Seismic Response of R. C. Setback Buildings: by Shahrooz, B.M. and Moehle, J.P., October 1987.

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UCB/EERC-88/03

UCB/EERC-88/04

UCB/EERC-87/17

UCB/EERC-87/18

UCB/EERC-87/19

UCB/EERC-88/06

UCB/EERC-88/07

UCB/EERC-88/08

UCB/EERC-88/09

UCB/EERC-87/20

UCB/EERC"87/21

UCB/EERC-87122

UCB/EERC-88/0 I

UCB/EERC-88/02

"The Effect of Slabs on the Flexural Behavior of Beams: by Pantazopoulou, S.J. and Moehle, J.P., October 1987, (PB90 262 7oo)A07.

"Design Procedure for R-FBI Bearings: by Mostaghel, N. and Kelly, J.~., November 1987, (PB90 262 718)A04.

"Analytical Models for Predicting the Lateral Response of R C Shear Walls: Evaluation of their Reliability," by Vulcano, A. and Ber­tero, V.V., November 1987.

"Earthquake Response of Torsionally-Coupled Buildings: by Hejal, R. and Chopra, A.K., December 1987.

"Dynamic Reservoir Interaction with Monticello Dam: by Clough, R.W., Ghanaat, Y. and Qiu, X-F., December 1987.

"Strength Evaluation of Coarse-Grained Soils: by Siddiqi, F.H., Seed, R.B., Chan, C.K., Seed, H.B. and Pyke, R.M., December 1987.

"Seismic Behavior of Concentrically Braced Steel Frames: by Khatib, I., Mahin, S.A. and Pister, K.S., January 1988.

"Experimental Evaluation of Seismic Isolation of Medium-Rise Structures Subject to Uplift: by Griffith, M.C., Kelly, J.M., Coveney,VA and Koh, e.G., January 1988.

"Cyclic Behavior of Steel Double Angle Connections," by Astaneh"AsI, A. and Nader, M.N., January 1988.

"Re-evaluation of the Slide in the Lower San Fernando Dam in the Earthquake of Feb. 9, 1971: by Seed, H.B., Seed, R.B., Harder,L.F. and Jong, H.·L., April 1988.

UCB/EERC"88/05 -Experimental Evaluation of Seismic Isolation of a Nine-Story Braced Steel Frame Subject to Uplift," by Griffith, M.e., Kelly, J.M. andAiken, I.D., May 1988.

"DRAIN-2DX User Guide.," by Allahabadi, R. and Powell, G.H., March 1988.

"Cylindrical Fluid Containers in Base-Isolated Structures: by Chalhoub, M.S. and Kelly, J.M. , April 1988.

"Analysis of Near·Source Waves: Separation of Wave Types using Strong Motion Array Recordings: by Darragh, R.B., June 1988.

"Alternatives to Standard Mode Superposition for Analysis of Non-Classically Damped Systems: by Kusainov, A.A. and Clough, R.W.,June 1988.

UCB/EERC-88/10 'The Landslide at the Port of Nice on October 16, 1979: by Seed, H.B., Seed, R.B., Schlosser, F., Blondeau, F. and Juran, I., June1988.

UCB/EERC-88/11

UCB/EERC-88/12

UCB/EERC-881l3

UCB/EERC-88/14

UCB/EERC-88/15

UCB/EERC-881l6

UCB/EERC·88/17

UCB/EERC-88/18

UCB/EERC-88/19

UCB/EERC-88/20

UCB/EERC-89/01

UCB/EERC-89/02

UCB/EERC-89/03

UCB/EERC-89/04

UCB/EERC-89/05

UCB/EERC-89/06

UCB/EERC-89/07

UCB/EERC-89/08

UCB/EERC-89/09

UCB/EERC·89/10

UCB/EERC-891l1

UCB/EERC-89/12

UCB/EERC-89/13

UCB/EERC-89/14

UCB/EERC-891l5

"Liquefaction Potential of Sand Deposits Under Low Levels of Excitation: by Carter, D.P. and Seed, H.B., August 1988.

"Nonlinear Analysis of Reinforced Concrete Frames Under Cyclic Load Reversals, - by Filippou, F.e. and Issa, A., September 1988.

"Implications of Recorded Earthquake Ground Motions on Seismic Design of Building Structures: by Uang, e.-M. and Bertero, V.V.,November 1988.

"An Experimental Study of the Behavior of Dual Steel Systems," by Whittaker, A.S. , Uang, C.-M. and Bertero, V.V., September 1988.

"Dynamic Moduli and Damping Ratios for Cohesive Soils: by Sun, J.I., Golesorkhi, R. and Seed, H.B., August 1988.

"Reinforced Concrete Flat Plates Under Lateral Load: An Experimental Study Including Biaxial Effects: by Pan, A. and Moehle, J.,October 1988.

"Earthquake Engineering Research at Berkeley - 1988: by EERC, November 1988.

"Use of Energy as a Design Criterion in Earthquake·Resistant Design: by Uang, e.-M. and Bertero, V.V., November 1988.

"Steel Beam-Column Joints in Seismic Moment Resisting Frames," by Tsai, K.-c. and Popov, E.P., November 1988.

"Base Isolation in Japan, 1988: by Kelly, J.M., December 1988.

-Behavior of Long Links in Eccentrically Braced Frames,- by Engelhardt, M.D. and Popov, E.P., January 1989.

"Earthquake Simulator Testing of Steel Plate Added Damping and Stiffness Elements: by Whittaker, A., Bertero, V.V., Alonso, J. andThompson, e., January 1989.

"Implications of Site Effects in the Mexico City Earthquake of Sept. 19, 1985 for Earthquake-Resistant Design Criteria in the San Fran­cisco Bay Area of California: by Seed, H.B. and Sun, J.I., March 1989.

-Earthquake Analysis and Response of Intake-outlet Towers: by Goyal, A. and Chopra, A.K., July 1989.

"The 1985 Chile Earthquake: An Evaluation of Structural Requirements for Bearing Wall Buildings: by Wallace, J.W. and Moehle,J.P., July 1989.

"Effects of Spatial Variation of Ground Motions on Large Multiply-Supported Structures: by Hao, H., July 1989.

-EADAP· Enhanced Arch Dam Analysis Program: Users's Manual: by Ghanaat, Y. and Clough, R.W., August 1989.

"Seismic Performance of Steel Moment Frames Plastically Designed by Least Squares Stress Fields,- by Ohi, K. and Mahin, S.A.,August 1989.

-Feasibility and Performance Studies on Improving the Earthquake Resistance of New and Existing Buildings Using the Friction Pendu­lum System: by zayas, V., Low, S., Mahin, SA. and Bozzo, L., July 1989.

"Measurement and Elimination of Membrane Compliance Effects in Undrained Triaxial Testing: by Nicholson, P.G., Seed. R.B. andAnwar, H., September 1989.

'Static Tilt Behavior of Unanchored Cylindrical Tanks: by Lau, D.T. and Clough, R.W., September 1989.

"ADAP-88: A Computer Program for Nonlinear Earthquake Analysis of Concrete Arch Dams: by Fenves, G.L., Mojtahedi, S. and Rei"mer, R.B., September 1989.

'Mechanics of Low Shape Factor Elastomeric Seismic Isolation Bearings,' by Aiken, 1.0., Kelly, J.M. and Tajirian, F., December 1989.

'Preliminary Report on the Seismological and Engineering Aspects of the October 17, 1989 Santa Cruz (Loma Prieta) Earthquake: byEERC, October 1989.

-Experimental Studies of a Single Story Steel Structure Tested with Fixed, Semi·Rigid and Flexible Connections: by Nader. M.N. andAstaneh-AsI, A., August 1989.

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UCBlEERC-89/16

UCB/EERC-90/0 I

UCB/EERC-90/02

UCB/EERC-90/03

UCB/EERC-90/04

UCB/EERC-90/05

UCB/EERC-90/06

UCB/EERC-90/07

UCB/EERC-90/08

UCB/EERC-90/09

UCB/EERC-90/10

UCB/EERC-90/11

UCB/EERC-90112

UCB/EERC-90/13

UCB/EERC-90/14

UCB/EERC-90/15

UCB/EERC-90/16

UCB/EERC-90/17

UCB/EERC-90118

UCB/EERC-90/19

UCB/EERC-90/20

UCB/EERC-90/21

UCB/EERC-91/01

UCB/EERC-91/02

UCB/EERC-91/03

UCB/EERC-91/04

UCB/EERC-91105

UCB/EERC-91/06

-Collapse of the Cypress Street Viaduct as a Result of the Loma Prieta Earthquake: by Nims, D.K., Miranda, E., Aiken, I.D.. Whit­taker, A.S. and Bertero, V.V., November 1989.

-Mechanics of High-Shape Factor Elastomeric Seismic Isolation Bearings," by Kelly, J.M., Aiken, LD. and Tajirian, F.F., March 1990.

"Javid's Paradox: The Influence of Preform on the Modes of Vibrating Beams: by Kelly, J.M., Sackman, J.L. and Javid, A., May 1990.

-Earthquake Simulator Testing and Analytical Studies of Two Energy-Absorbing Systems for Multistory Structures," by Aiken, LD. andKelly, J.M., October 1990.

"Damage to the San Francisco-Oakland Bay Bridge During the October 17,1989 Earthquake: by Astaneh, A., June 1990.

"Preliminary Report on the Principal Geotechnical Aspects of the October 17, 1989 Loma Prieta Earthquake," by Seed, R.B., Dicken­son, S.E., Riemer, M.F., Bray, J.D., Sitar, N., Mitchell, J.K., Idriss, LM., Kayen, R.E., Kropp, A., Harder, L.F., Jr. and Power, M.S.,April 1990.

"Models of Critical Regions in Reinforced Concrete Frames Under Seismic Excitations: by Zulfiqar, N. and Filippou, F., May 1990.

-A Unified Earthquake-Resistant Design Method for Steel Frames Using ARMA Models: by Takewaici, I., Conte, J.P., Mahin, S.A. andPister, K.S., June 1990.

"Soil Conditions and Earthquake Hazard Mitigation in the Marina District of San Francisco: by Mitchell, J.K., Masood, T., Kayen,R.E. and Seed, R.B., May 1990.

"Influence of the Earthquake Ground Motion Process and Structural Properties on Response Characteristics of Simple Structures: byConte, J.P., Pister, K.S. and Mahin, S.A., July 1990.

"Experimental Testing of the Resilient-Friction Base Isolation System: by Clark, P.W. and Kelly, J.M., July 1990.

-Seismic Hazard Analysis: Improved Models, Uncertainties and Sensitivities," by Araya, R. and Der Kiureghian, A., March 1988.

"Effects of Torsion on the Linear and Nonlinear Seismic Response of Structures: by Sedarat, H. and Bertero, V.V., September 1989.

-The Effects of Tectonic Movements on Stresses and Deformations in Earth Embankments: by Bray, J. D., Seed, R. B. and Seed, H. B.,September 1989.

-Inelastic Seismic Response of One-Story, Asymmetric-Plan Systems: by Geel, R.K. and Chopra, A.K., October 1990.

-Dynamic Crack Propagation: A Model for Near-Field Ground Motion.: by Seyyedian, H. and Kelly, J.M., 1990.

"SenSitivity of Long-Period Response Spectra to System Initial Conditions: by Blasquez, R., Ventura, C. and Kelly, J.M., 1990.

"Behavior of Peak Values and Spectral Ordinates of Near-Source Strong Ground-Motion over a Dense Array: by Niazi, M., June 1990.

"Material Characterization of Elastomers used in Earthquake Base Isolation: by Papoulia, K.D. and Kelly, J.M., 1990.

-Cyclic Behavior of Steel Top-and-Bottom Plate Moment Connections: by Harriott, J.D. and Astaneh, A., August 1990.

"Seismic Response Evaluation of an Instrumented Six Story Steel Building," by Shen, J.-H. and Astaneh, A., December 1990.

"Observations and Implications of Tests on the Cypress Street Viaduct Test Structure: by Mahin, S.A., Moehle, J.P., Stephen. R.M.,Bolio. M. and Qi, X., December 1990.

-Experimental Evaluation of Nitinol for Energy Dissipation in Structures: by Nims, D.K., Sasaki, K.K. and Kelly, J.M., 1991.

"Displacement Design Approach for Reinforced Concrete Structures Subjected to Earthquakes: by Qi, X. and Moehle, J.P.. January1991.

-Shake Table Tests of Long Period Isolation System for Nuclear Facilities at Soft Soil Sites, by Kelly, J.M., March 1991.

-Dynamic and Failure Characteristics of Bridgestone Isolation Bearings: by Kelly, J.M., April 1991.

"Base Sliding Response of Concrete Gravity Dams to Earthquakes: by Chopra, A.K. and Zhang, L.. May 1991.

"Computation of Spatially Varying Ground Motion and Foundation-Rock Impedance Matrices for Seismic Analysis of Arch Dams: byZhang, L. and Chopra, A.K., May 1991.

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